sfepy.terms.terms_dg module¶

Discontinous Galekrin method specific terms

Note¶

In einsum calls the following convention is used:

i represents iterating over all cells of a region;

n represents iterating over selected cells of a region, for example over cells on boundary;

b represents iterating over basis functions of state variable;

d represents iterating over basis functions of test variable;

k, l , m represent iterating over geometric dimensions, for example coordinates of velocity or facet normal vector or rows and columns of diffusion tensor;

q represents iterating over quadrature points;

f represents iterating over facets of cell;

class sfepy.terms.terms_dg.AdvectionDGFluxTerm(name, arg_str, integral, region, **kwargs)[source]¶

Lax-Friedrichs flux term for advection of scalar quantity $p$ with the advection velocity $\ul{a}$ given as a material parameter (a known function of space and time).

Definition:

$\int_{\partial{T_K}} \ul{n} \cdot \ul{f}^{*} (p_{in}, p_{out})q$

where

$\ul{f}^{*}(p_{in}, p_{out}) = \ul{a} \frac{p_{in} + p_{out}}{2} + (1 - \alpha) \ul{n} C \frac{ p_{in} - p_{out}}{2},$

$\alpha \in [0, 1]$ ; $\alpha = 0$ for upwind scheme, $\alpha = 1$ for central scheme, and

$C = \max_{p \in [?, ?]}\left\lvert n_x a_1 + n_y a_2 \right\rvert = \max_{p \in [?, ?]} \left\lvert \ul{n} \cdot \ul{a} \right\rvert$

the $p_{in}$ resp. $p_{out}$ is solution on the boundary of the element provided by element itself resp. its neighbor and $\ul{a}$ is advection velocity.

Call signature:

dw_dg_advect_laxfrie_flux

(opt_material, material_advelo, virtual, state)

Arguments 1:

material : $\ul{a}$
virtual : $q$
state : $p$

Arguments 3:

material : $\ul{a}$
virtual : $q$
state : $p$
opt_material : $\alpha$

alpha = 0¶

arg_shapes = [{'material_advelo': 'D, 1', 'opt_material': '.: 1', 'state': 1, 'virtual': (1, 'state')}, {'opt_material': None}]¶

arg_types = ('opt_material', 'material_advelo', 'virtual', 'state')¶

function(out, state, diff_var, field, region, advelo)[source]¶

get_fargs(alpha, advelo, test, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'cell'¶

modes = ('weak',)¶

name = 'dw_dg_advect_laxfrie_flux'¶

symbolic = {'expression': 'div(a*p)*w', 'map': {'a': 'material', 'p': 'state', 'v': 'virtual'}}¶

class sfepy.terms.terms_dg.DGTerm(name, arg_str, integral, region, **kwargs)[source]¶

Abstract base class for DG terms, provides alternative call_function and eval_real methods to accommodate returning iels and vals.

call_function(out, fargs)[source]¶

eval_real(shape, fargs, mode='eval', term_mode=None, diff_var=None, **kwargs)[source]¶

poly_space_basis = 'legendre'¶

class sfepy.terms.terms_dg.DiffusionDGFluxTerm(name, arg_str, integral, region, **kwargs)[source]¶

Basic DG diffusion flux term for scalar quantity.

Definition:

$\int_{\partial{T_K}} D \langle \nabla p \rangle [q] \mbox{ , } \int_{\partial{T_K}} D \langle \nabla q \rangle [p]$

where

$\langle \nabla \phi \rangle = \frac{\nabla\phi_{in} + \nabla\phi_{out}}{2}$

$[\phi] = \phi_{in} - \phi_{out}$

Math:

The $p_{in}$ resp. $p_{out}$ is solution on the boundary of the element provided by element itself resp. its neighbour.

Call signature:

dw_dg_diffusion_flux	`(material, state, virtual)`
	`(material, virtual, state)`

Arguments 1:

material : $D$
state : $p$
virtual : $q$

Arguments 2:

material : $D$
virtual : $q$
state : $p$

arg_shapes = [{'material': '1, 1', 'state/avg_state': 1, 'state/avg_virtual': 1, 'virtual/avg_state': (1, None), 'virtual/avg_virtual': (1, None)}]¶

arg_types = (('material', 'state', 'virtual'), ('material', 'virtual', 'state'))¶

function(out, state, diff_var, field, region, D)[source]¶

get_fargs(diff_tensor, test, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'cell'¶

modes = ('avg_state', 'avg_virtual')¶

name = 'dw_dg_diffusion_flux'¶

class sfepy.terms.terms_dg.DiffusionInteriorPenaltyTerm(name, arg_str, integral, region, **kwargs)[source]¶

Penalty term used to counteract discontinuity arising when modeling diffusion using Discontinuous Galerkin schemes.

Definition:

$\int_{\partial{T_K}} \bar{D} C_w \frac{Ord^2}{d(\partial{T_K})}[p][q]$

where

$[\phi] = \phi_{in} - \phi_{out}$

Math:

the $p_{in}$ resp. $p_{out}$ is solution on the boundary of the element provided by element itself resp. its neighbour.

Call signature:

dw_dg_interior_penalty

(material, material_Cw, virtual, state)

Arguments:

material : $D$
material : $C_w$
state : $p$
virtual : $q$

arg_shapes = [{'material': '1, 1', 'material_Cw': '.: 1', 'state': 1, 'virtual': (1, 'state')}]¶

arg_types = ('material', 'material_Cw', 'virtual', 'state')¶

function(out, state, diff_var, field, region, Cw, diff_tensor)[source]¶

get_fargs(diff_tensor, Cw, test, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('weak',)¶

name = 'dw_dg_interior_penalty'¶

class sfepy.terms.terms_dg.NonlinearHyperbolicDGFluxTerm(name, arg_str, integral, region, **kwargs)[source]¶

Lax-Friedrichs flux term for nonlinear hyperpolic term of scalar quantity $p$ with the vector function $\ul{f}$ given as a material parameter.

Definition:

$\int_{\partial{T_K}} \ul{n} \cdot f^{*} (p_{in}, p_{out})q$

where

$\ul{f}^{*}(p_{in}, p_{out}) = \frac{\ul{f}(p_{in}) + \ul{f}(p_{out})}{2} + (1 - \alpha) \ul{n} C \frac{ p_{in} - p_{out}}{2},$

$\alpha \in [0, 1]$ ; $\alpha = 0$ for upwind scheme, $\alpha = 1$ for central scheme, and

$C = \max_{p \in [?, ?]}\left\lvert n_x \frac{d f_1}{d p} + n_y \frac{d f_2}{d p} + \cdots \right\rvert = \max_{p \in [?, ?]} \left\lvert \vec{n}\cdot\frac{d\ul{f}}{dp}(p) \right\rvert$

the $p_{in}$ resp. $p_{out}$ is solution on the boundary of the element provided by element itself resp. its neighbor.

Call signature:

dw_dg_nonlinear_laxfrie_flux

(opt_material, fun, fun_d, virtual, state)

Arguments 1:

material : $\ul{f}$
material : $\frac{d\ul{f}}{d p}$
virtual : $q$
state : $p$

Arguments 3:

material : $\ul{f}$
material : $\frac{d\ul{f}}{d p}$
virtual : $q$
state : $p$
opt_material : $\alpha$

alf = 0¶

arg_shapes = [{'material_fun': '.: 1', 'material_fun_d': '.: 1', 'opt_material': '.: 1', 'state': 1, 'virtual': (1, 'state')}, {'opt_material': None}]¶

arg_types = ('opt_material', 'fun', 'fun_d', 'virtual', 'state')¶

function(out, state, field, region, f, df)[source]¶

get_fargs(alpha, fun, dfun, test, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'cell'¶

modes = ('weak',)¶

name = 'dw_dg_nonlinear_laxfrie_flux'¶

symbolic = {'expression': 'div(f(p))*w', 'map': {'f': 'function', 'p': 'state', 'v': 'virtual'}}¶

class sfepy.terms.terms_dg.NonlinearScalarDotGradTerm(name, arg_str, integral, region, **kwargs)[source]¶

Product of virtual and divergence of vector function of state or volume dot product of vector function of state and gradient of scalar virtual.

Definition:

$\int_{\Omega} q \cdot \nabla \cdot \ul{f}(p) = \int_{\Omega} q \cdot \text{div} \ul{f}(p) \mbox{ , } \int_{\Omega} \ul{f}(p) \cdot \nabla q$

Call signature:

dw_ns_dot_grad_s	`(fun, fun_d, virtual, state)`
	`(fun, fun_d, state, virtual)`

Arguments 1:

function : $\ul{f}$
virtual : $q$
state : $p$

Arguments 2:

function : $\ul{f}$
state : $p$
virtual : $q$

TODO maybe this term would fit better to terms_dot?

arg_shapes = [{'material_fun': '.: 1', 'material_fun_d': '.: 1', 'state/grad_state': 1, 'state/grad_virtual': 1, 'virtual/grad_state': (1, None), 'virtual/grad_virtual': (1, None)}]¶

arg_types = (('fun', 'fun_d', 'virtual', 'state'), ('fun', 'fun_d', 'state', 'virtual'))¶

static function(out, out_qp, geo, fmode)[source]¶

get_fargs(fun, dfun, var1, var2, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = ('grad_state', 'grad_virtual')¶

name = 'dw_ns_dot_grad_s'¶