Detailed Description

A hierarchical cubegrid of arbitrary dimension.

This is an axis parallel hexaedral grid, with non-conform refinement structure. All vertices are explicitly generated and stored. Only practical for low dimensions, i.e. 1 to 4 or perhaps 5.

Definition at line 17 of file cubegrid.m.

Inheritance diagram for cubegrid:

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Collaboration diagram for cubegrid:

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Public Member Functions
	cubegrid (varargin)
	constructor of a hierarchical cubegrid More...

function [ compress , new ] =	tpart_refine (lids, new_midpoints)
	refines certain grid entities More...

function res =	check_consistency ()

function leaf_element =	coord2leaf_element (coord)
	function giving the leaf_element of a grid where the values of the coord-vector can be found (coord- can for example be a set of mu-values in a parameter grid. This function utilizes a \(O(log(n))\)-search algorithm. More...

function	demo ()
	small script demonstrating the possibilities of the cubegrid class.

function	display ()
	cubegrid's display method. More...

function ret =	get (propertyname, varargin)
	get-method for cubegrids More...

function [ p0 , p1 ] =	get_edges (gid)
	method determining the world coordinates of the edges of a single element with global index `gid`. More...

function gids =	get_leafgids ()
	return leaf ids of cubegrid, i.e. vector with global element indices of leaf elements More...

function range =	get_ranges_of_element (element)
	return the ranges corresponding to the element of the grid, i.e. maximum vertex range of the part_num element of the grid. More...

function vols =	get_volume (gids)
	method determining the volumes of a set of elements More...

function lids =	gid2lid (gids)
	function converting global element ids to leaf element ids More...

function gids =	lid2gid (lids)
	function converting leaf-element ids to global element ids More...

function p =	plot (params)
	plot method for a cubegrid. More...

function p =	plot_grid (params)
	plot method for a cubegrid. A line plot is performed. Text can be plotted in the leafs of the grid. More...

function p =	plot_leafelement_data (data, params)
	plot method for piecewise constant data on the leaf level of the grid. More...

function p =	plot_leafvertex_data (data, params)
	plot method for data on the leaf level of the grid. More...

function ngrid =	refine (lids)
	refinement routine for cubegrid. More...

function ngrid =	remove_duplicate_vertices (epsilon)
	method to be used, if a vertex list in the grid should be compressed More...

function cubegrid gcopy =	copy ()
	a deep copy of the cubegrid More...

Public Member Functions inherited from gridbase
	gridbase ()
	default constructor for gridbase

function res =	check_consistency ()
	function checking the consistency of a polygonal grid, i.e. checking, whether the edge centroids correspond (implicit neighbour-index test) More...

function	display ()
	default implementation for display method for unstructured polygonal grids. is used by rectgrid and triagrid

function F =	edge_quad_eval (elids, edgeids, degree, FF)
	Compute an edge integral of a scalar function in various edges simultaneously. Approximation by Gauss-quadratures are performed. More...

function F =	edge_quad_eval_mean (elids, edgeids, degree, FF)
	Compute an edge-average integral of a scalar function in various edges simultaneously. Approximation by Gauss-quadratures are performed. More...

function PP =	edge_quad_points (elids, edgeids, degree)
	get the evaluation points for a quadrature over edges of the given grid. More...

function [ P1 , P2 ] =	get_edge_points (elids, edgeids)
	function extracting edge coordinates from the grid. More...

function ENBI =	get_enbi (edge, tstep)
	function assembling a matrix with the 5 neighbour's cell indices that are needed in order to compute the gradient over the edge given by `edge` in each row. see also the sketch below. More...

function gridp =	gridpart (eind)
	function extracting a part of a triagrid or rectgrid defined by the given element indices in the vector `eind`. More...

function	inspect (params)
	function plotting various properties of the current grid. More...

function p =	plot_polygon_grid (params)
	plot method for a 2D polygonal grid. This routine can be used for triangular and rectangular grids. More...

function p =	plot_element_data (data, plot_params)
	plot method for a 2d polygonal grid and elementwise constant data routine can be used for triangular and rectangular grids. More...

function	plot_element_data_sequence (data, plot_params)
	plot of a sequence of element_data on the given grid (constructed if empty). performs simple call of plot_data_sequence. More...

function p =	plot_vertex_data (data, params)
	plot method for vertex data on a 2D polygonal grid. More...

function gridbase grid =	set_boundary_types (params)
	function setting the boundary types of a polygonal grid. More...

Public Attributes
	dimension
	dimension of grid/number of coordinates per vertex

	vertex
	`vertex(i,j)` is the j-th coordinate of i-th vertex point

	vertexindex
	`vertexindex(id,j)` is the index of j-th vertex of element `id`

	level
	`level(id)` equals refinement level of element `id`

	isleaf
	`isleaf(id)` indicates, wether element `id` is leaf or not

	firstchild
	`firstchild(id)` indicates the lowest-index of a child element

	refine_steps
	number of refinement-steps, that have been performed on the grid so far

	creation_step
	`creation_step(id)` stores the refinement step number that has led to the element `id`.

Public Attributes inherited from gridbase
	nelements
	number of overall elements (leaf + nonleaf)

	nvertices
	number of vertices

	nneigh
	number of neighbor elements

	A
	vector of element area

	Ainv
	vector of inverted element area

	VI
	matrix of vertex indices: `VI(i,j)` is the global index of `j`-th vertex of element `i`

	X
	vector of vertex x-coordinates

	Y
	vector of vertex y-coordinates

	CX
	vector of centroid x-values

	CY
	vector of centroid y-values

	NBI
	`NBI(i,j)` = element index of `j`-th neighbour of element `i` boundary faces are set to -1 or negative values are requested by `params.boundary_type`

	INB
	`INB(i,j)` = local edge number in `NBI(i,j)` leading from element `NBI(i,j)` to element `i`, i.e. satisfying `NBI(NBI(i,j), INB(i,j)) = i`

	EL
	`EL(i,j)` = length of edge from element `i` to neighbour `j`

	DC
	`DC(i,j)` = distance from centroid of element `i` to NB `j` for boundary elements, this is the distance to the reflected element (for use in boundary treatment)

	NX
	`NX(i,j)` = x-coordinate of unit outer normal of edge from el `i` to NB `j`

	NY
	`NY(i,j)` = y-coordinate of unit outer normal of edge from el `i` to NB `j`

	ECX
	`ECX(i,j)` = x-coordinate of midpoint of edge from el `i` to NB `j`

	ECY
	`ECY(i,j)` = x-coordinate of midpoint of edge from el `i` to NB `j`

	SX
	vector of x-coordinates of point \(S_i\) (for rect: identical to centroids) More...

	SY
	vector of y-coordinates of point \(S_i\) (for rect: identical to centroids) More...

	ESX
	`ESX(i,j)` = x-coordinate of point \(S_ij\) on edge el `i` to NB `j`

	ESY
	`ESY(i,j)` = x-coordinate of point \(S_ij\) on edge el `i` to NB `j`

	DS
	`DS(i,j)` = distance from \(S_i\) of element `i` to \(S_j\) of NB `j` for boundary elements, this is the distance to the reflected element (for use in boundary treatment)

	hmin
	minimal element-diameter

	alpha
	geometry bound (simultaneously satisfying \[\alpha h_i^d \leq A(T_i),\] \[\alpha \text{diam}(T_i) \leq h_i^{d-1}\] and \[\alpha h_i \leq \text{distance(midpoint i to any neighbour)}\]

	JIT
	Jacobian inverse transposed `JIT(i,:,:)` is a `2x2`-matrix of the Jacobian Inverse Transposed on element `i`

Constructor & Destructor Documentation

cubegrid.cubegrid ( varargin )

constructor of a hierarchical cubegrid

Preliminaries

In the following

gid denotes global element indices, i.e. also covering non-leaf elements,
lid denotes indices for leaf indices.

Conversion can be performed by gids = lid2gid(grid,lid) Conversion can be performed by lids = gid2lid(grid,gid)

The constructor has the following

Synopsis

cubegrid() : construction of a default cubegrid (2d unit square)
cubegrid(cgrid) : copy-constructor
cubegrid(params): generate cubegrid with certain options. The
argument params requires the following fields
1. range : cell array of intervals, where the array lengths determine the grid dimension
2. numintervals: vector with number of intervals per dimension
perhaps later: constructor by duneDGF-file?

Parameters

varargin varargin

Generated fields of grid:

dimension — dimension of grid/number of coordinates per vertex
nelements — number of overall elements (leaf + nonleaf);
nvertices — number of vertices;
vertex — vertex(i,j) = j-th coordinate of i-th vertex point
vertexindex — vertexindex(id,j) = index of j-th vertex of element id
level — level(id) = level, i.e. refinement steps of element id
isleaf — isleaf(id) = 0 or 1 whether element id is leaf or not
refine_steps — number of refinement-steps, that have been performed on the grid so far
creation_step — creation_step(id) stores the refinement step number that has led to the element id.

Definition at line 98 of file cubegrid.m.

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Member Function Documentation

function leaf_element = cubegrid.coord2leaf_element ( coord )

function giving the leaf_element of a grid where the values of the coord-vector can be found (coord- can for example be a set of mu-values in a parameter grid. This function utilizes a \(O(log(n))\)-search algorithm.

coord : vector of coord-values (must have same dimension as grid)

15.02.2010 Markus Dihlmann

Parameters

coord coord

Return values

leaf_element leaf element

Definition at line 18 of file coord2leaf_element.m.

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function cubegrid gcopy = cubegrid.copy ( )

virtual

a deep copy of the cubegrid

Return values

gcopy a deep copy of this instance also

Implements gridbase.

Definition at line 334 of file cubegrid.m.

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function cubegrid.display ( )

cubegrid's display method.

All scalar quantities are displayed. Internal pointlists, vectors and flags of elements are not displayed. These can be accessed by the get() method.

Definition at line 18 of file display.m.

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function ret = cubegrid.get	(	propertyname,
		varargin
	)

get-method for cubegrids

Parameters

propertyname

A property name of the grid which can be one of the following

level : get vector of levels of all elements
dimension : get dimension of grid
nvertices : get number of vertices of grid
nelements : get number of elements of grid
isleaf : get vector of isleaf-indices of all elements
vertexindex returns a matrix (i,j) index of j-th vertex of element i a further optional argument with element indices can be passed, then only the vertices of these elements are returned,
get(grid,'vertexindex',[2,3])

returns a 2 x nvertices_per_element matrix with indices
vertex : returns a matrix of all vertex coordinates, i.e. ret(i,j) => j-th coordinate of i-th vertex point optionally, a subset of vertex-indices can be determined as further parameter, i.e.
get(grid,'vertex',[2 3 4])

produces a 3 x dim matrix with the coordinates of the points

or property can be equal to one of the following strings returning quantities which are derived from the above grid properties

nleafelements : get number of leaf elements of grid
leafelements : get vector of indices of elements that are leaf elements
leafcogs : returns a matrix of all centers of gravity of the leaf elements. ret(i,j) is the j-th coordinate of the i-th leaf element

varargin An optional list of arguments as described above.

Return values

ret ret

Definition at line 18 of file get.m.

function [ p0 , p1 ] = cubegrid.get_edges ( gid )

method determining the world coordinates of the edges of a single element with global index gid.

i-th edge goes from p0(:,i) to p1(:,i)

Note: vectorized version for multiple elements not yet implemented

Parameters

gid	global index of element for which edge coordinates are requested

Return values

p0	lowest coordinate of the edge
p1	highest coordinate of the edge

Definition at line 18 of file get_edges.m.

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function gids = cubegrid.get_leafgids ( )

return leaf ids of cubegrid, i.e. vector with global element indices of leaf elements

Bernard Haasdonk 27.3.2007

Return values

gids	gids
vector	of global leaf indices

Definition at line 18 of file get_leafgids.m.

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function range = cubegrid.get_ranges_of_element ( element )

return the ranges corresponding to the element of the grid, i.e. maximum vertex range of the part_num element of the grid.

Markus Dihlmann 16.02.2010

Parameters

element element

Return values

range range

Definition at line 18 of file get_ranges_of_element.m.

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function vols = cubegrid.get_volume ( gids )

method determining the volumes of a set of elements

A simple product of edge-lengths is computed, i.e. the edges are assumed to be axis-parallel.

Note: The computation for multiple gids is trivially performed by a loop. Vectorization must be done sometime!!

Parameters

gids	Global indices

Return values

vols	The volumes for the elements given by `gids`.

Definition at line 18 of file get_volume.m.

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function lids = cubegrid.gid2lid ( gids )

function converting global element ids to leaf element ids

Parameters

gids	global indices of elements to which the local indices shall be computed.

Return values

lids	leaf element ids or `-1` if a global id was not a leaf element

Definition at line 18 of file gid2lid.m.

function gids = cubegrid.lid2gid ( lids )

function converting leaf-element ids to global element ids

Parameters

lids	local indices for which the global ones shall be computed

Return values

gids	global indices of leaf indices

Definition at line 18 of file lid2gid.m.

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function p = cubegrid.plot ( params )

plot method for a cubegrid.

A line plot is performed. Currently implementation for more than 3D not implemented, as this would be a line-mess

Bernard Haasdonk 1.3.2007

Parameters

params params

Return values

p	this is the list of handles to the graphics primitives

Required fields of params:

LineStyle — LineStyle

Optional fields of params:

color — RGB vector
shrink_factor — if this flag is given, the elements are plotted shrinked
plot_level — if this flag is nonzero, a level plot is performed
level — this integer must be specified in case of level-plot
plot_patch — if this flag is set (only 2D and 3D) the plot is done by colored patches
axis_equal — if this flag is set, set axis to equal scale

Definition at line 18 of file plot.m.

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function p = cubegrid.plot_grid ( params )

plot method for a cubegrid. A line plot is performed. Text can be plotted in the leafs of the grid.

p is the list of handles to the graphics primitives

optional params: color : RGB vector shrink_factor: if this flag is given, the elements are plotted shrinked plot_level: if this flag is nonzero, a level plot is performed level : this integer must be specified in case of level-plot plot_patch : if this flag is set (only 2D and 3D) the plot is done by colored patches axis_equal : if this flag is set, set axis to equal scale

leaftext : vector of strings. Length must be equal to number of leafelements. The text is shown in the cogs (center of gravities) of the leafelements.

Markus Dihlmann 18.02.2010

Parameters

params params

Return values

p p

Required fields of params:

text — text

Definition at line 18 of file plot_grid.m.

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function p = cubegrid.plot_leafelement_data	(	data,
		params
	)

plot method for piecewise constant data on the leaf level of the grid.

Currently only implemented for 2D. A patch plot is performed.

See also: plot_element_data() for general possible options in params

Parameters

data	data
params	params

Return values

p	this is the list of handles to the graphics primitives

Definition at line 18 of file plot_leafelement_data.m.

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function p = cubegrid.plot_leafvertex_data	(	data,
		params
	)

plot method for data on the leaf level of the grid.

Currently only implemented for 2D. A patch plot is performed, such that the vertices have corresponding values as given by data, the values in between are interpolated by MATLAB.

See also: plot_vertex_data() for optional params fields

Parameters

data	data
params	params

Return values

p	this is the list of handles to the graphics primitives

Definition at line 18 of file plot_leafvertex_data.m.

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function ngrid = cubegrid.refine ( lids )

refinement routine for cubegrid.

Refinement rule is a simple division into \(2^dim\) smaller cubes After refinement, a condensation of the vertex list is performed by remove_duplicate_vetrices(), i.e. duplicate vertices generated during refinement are removed.

Parameters

lids	List of leaf-element ids of the to-be-refined codim `0` entities. Only leaf-elements can reasonably be refined, so do not pass global elements here, as internally a translation into global element ids is performed.

Return values

ngrid the refined grid

Definition at line 18 of file refine.m.

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function ngrid = cubegrid.remove_duplicate_vertices ( epsilon )

method to be used, if a vertex list in the grid should be compressed

This might be required after refinement of a grid, merging of grids, etc. Duplicate detection is based on l2-error deviation thresholded by epsilon.

Parameters

epsilon l2-error deviation threshold. (Default = 1e-6)

Return values

ngrid the compressed grid

Generated fields of ngrid: