Solve the within-group transport equation. More...

Inheritance diagram for InnerIteration:

Collaboration diagram for InnerIteration:

Public Member Functions
function	InnerIteration ()
	Class constructor.
function	reset_convergence (in this, in max_iters, in tolerance)
	Solve the within-group problem.
function	display_me (in this)
Protected Member Functions
function	setup_base (in this, in)
	Setup the base solver.
function	initialize_scatter (in this)
	Prebuild scattering matrix for each cell.
function	build_scatter_source (in this, in g, in phi)
	Build the within-group scattering source.
function	build_all_scatter_source (in this, in g, in phi)
	Build all the scattering sources.
function	build_fixed_source (in this, in g)
	Build source for this group excluding within-group scatter.
function	build_external_source (in this, in g)
function	check_convergence (in this, in iteration, in flux_error)
function	print_iteration (in this, in it, in e0, in e1, in e2)
Protected Attributes
Property	d_input
	User input.
Property	d_state
	State vectors.
Property	d_boundary
	Boundary fluxes.
Property	d_mesh
	Problem mesh (either Cartesian mesh or MOC tracking)
Property	d_mat
	Materials definitions.
Property	d_quadrature
	Angular mesh.
Property	d_equation
	Spatial discretization.
Property	d_sweeper
	Sweeper over the space-angle domain.
Property	d_external_source
	User-defined external source.
Property	d_fission_source
	Fission source, if used.
Property	d_fixed_source
	Any source that remains "fixed" within the group solve.
Property	d_scatter_source
	The within group scattering source.
Property	d_sigma_s
	Scattering cross-section vector for faster source computation.
Property	d_max_iters
Property	d_tolerance
Property	d_g
Property	d_M
	Moments to discrete operator.

Detailed Description

Solve the within-group transport equation.

The within-group transport equation in operator form is

$\mathbf{L}\psi = \mathbf{MS}\phi + Q$

where $\mathbf{L}$ is the streaming and collision operator, {M} is the moment-to-discrete operator, {S} is the scattering operator, and Q represents any source considered fixed, which includes in-scatter, fission, and external sources.

What we are really after is the scalar flux and possibly its higher order moments. Consequently, we are able to solve a somewhat different problem then the within group transport equation above. Let us operate on both sides by $\mathbf{L}^{-1}$ followed by $\mathbf{D}$ to get

$(\mathbf{I} - \mathbf{D}\mathbf{L}^{-1}\mathbf{MS})\phi = \mathbf{D} \mathbf{L}^{-1} Q \, .$

Here, $\mathbf{D}$ is the discrete-to-moment operator, defined such that $\phi = \mathbf{D}\psi$ .

Notice this is nothing but a linear system of the form $\mathbf{A}x = b$ where

$\mathbf{A} = (\mathbf{I} - \mathbf{D}\mathbf{L}^{-1}\mathbf{MS})$

and

$b = \mathbf{D} \mathbf{L}^{-1} Q \, .$

Moreover, $ b$ is just the uncollided flux.

A nice overview of approaches for this inner iteration is given by Larsen and Morel in Nuclear Computational Science . We have implement the standard source iteration method, Livolant acceleration (an extrapolation technique), and solvers that use MATLAB's own GMRES (and other Krylov solvers).

Input parameters specific to InnerIteration and derived classes:

inner_max_iters (default: 100)
inner_tolerance (default: 1e-5)

See also:: SourceIteration, Livolant, GMRESIteration

Constructor & Destructor Documentation

function InnerIteration ( )

Class constructor.

Parameters:

input	Input database.
state	State vectors, etc.
boundary	Boundary fluxes.
mesh	Problem mesh.
mat	Material definitions.
quadrature	Angular mesh..
external_source	User-defined external source.
fission_source	Fission source.

Returns:: Instance of the InnerIteration class.

Member Function Documentation

function build_all_scatter_source	(	in	this,
		in	g,
		in	phi
	)		`[protected]`

Build all the scattering sources.

Parameters:

g	Group for this problem. (I.e. row in MG).
phi	Current MG flux.

function build_external_source	(	in	this,
		in	g
	)		`[protected]`

function build_fixed_source	(	in	this,
		in	g
	)		`[protected]`

Build source for this group excluding within-group scatter.

Parameters:

g	Group for this problem.

function build_scatter_source	(	in	this,
		in	g,
		in	phi
	)		`[protected]`

Build the within-group scattering source.

Parameters:

g	Group for this problem.
phi	Current group flux.

function check_convergence	(	in	this,
		in	iteration,
		in	flux_error
	)		`[protected]`

function display_me ( in this )

function initialize_scatter ( in this ) [protected]

Prebuild scattering matrix for each cell.

While not the most *memory* efficient, this saves *time* by eliminate lots of loops.

function print_iteration	(	in	this,
		in	it,
		in	e0,
		in	e1,
		in	e2
	)		`[protected]`

function reset_convergence	(	in	this,
		in	max_iters,
		in	tolerance
	)

Solve the within-group problem.

Parameters:

g	Group of the problem to be solved.

Returns:: Flux residual (L-inf norm) and iteration count. Reset convergence criteria. It can be useful to use a dynamically changing convergence criteria, especially for eigenproblems. For regular power iteration, it is wasteful to use tight tolerances on the inners when the source is not as tightly known.

Parameters:

max_iters	Maximum number of iterations.
tolerance	Maximum point-wise error in flux

function setup_base	(	in	this,
		in
	)		`[protected]`

Setup the base solver.

By keeping everything out of the constructor, we can avoid having to copy the signature for all derived classes, both in their definitions and whereever thisects are instantiated. However, derived classes have other setup issues that need to be done, and so this serves as a general setup function that must be called before other setup stuff.

Parameters:

input	Input database.
state	State vectors, etc.
boundary	Boundary fluxes.
mesh	Problem mesh.
mat	Material definitions.
quadrature	Angular mesh..
external_source	User-defined external source.
fission_source	Fission source.

Member Data Documentation

Property d_boundary [protected]

Boundary fluxes.

Property d_equation [protected]

Spatial discretization.

Property d_external_source [protected]

User-defined external source.

Property d_fission_source [protected]

Fission source, if used.

Property d_fixed_source [protected]

Any source that remains "fixed" within the group solve.

Property d_g [protected]

Property d_input [protected]

User input.

Property d_M [protected]

Moments to discrete operator.

Property d_mat [protected]

Materials definitions.

Property d_max_iters [protected]

Property d_mesh [protected]

Problem mesh (either Cartesian mesh or MOC tracking)

Property d_quadrature [protected]

Angular mesh.

Property d_scatter_source [protected]

The within group scattering source.

Property d_sigma_s [protected]

Scattering cross-section vector for faster source computation.

Property d_state [protected]

State vectors.

Property d_sweeper [protected]

Sweeper over the space-angle domain.

Property d_tolerance [protected]

The documentation for this class was generated from the following file:

/home/robertsj/Research/matlab_transport_pack/source/InnerIteration.m

InnerIteration Class Reference

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation