rectgrid Class Reference

Detailed Description

A cartesian rectangular grid in two dimensions with axis parallel elements.

General geometrical information is stored including neighbour information. Therefore also boundary neighbour relations can be specified. The boundary type is set to be symmetric by default. Additionally, "rectangles" can be defined: the edges with midpoints within such a rectangle are marked accordingly. By this, boundary edges can be marked for later special use Much (partially redundant) information is stored in the grid, which might be useful in simulations.

• Sorting of vertices is always counterclockwise SE,NE,NW,SW
• A local edge \(j \in \{1,\dots,4\}\) is connecting points \(j\) and \((j+1) \mod 4\)

Definition at line 17 of file rectgrid.m.

Inheritance diagram for rectgrid:

Collaboration diagram for rectgrid:

Public Member Functions
	rectgrid (varargin)
	constructor of a cartesian rectangular grid in 2 dimensions with axis parallel elements. More...
function	demo ()
	small script demonstrating the possibilities of the rectgrid class.
function p =	plot (params)
	plot of a rectgrid via plot_polygon_grid() More...
function res =	isequal (grid2)
	function comparing two grids field by field More...
function gridbase grid =	set_nbi (nbind, values)
	function setting some neighbour indices of a grid to specified values. More...
function gcopy =	copy ()
	returns a deep copy object of the grid implementation More...
function glob =	local2global (einds, loc, params)
	function performing a local to global coordinate change of vectors of coordinate pairs. 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...

Additional Inherited Members
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

rectgrid.rectgrid

(

varargin

)

constructor of a cartesian rectangular grid in 2 dimensions with axis parallel elements.

The constructor has sthe following

Synopsis

• rectgrid() : construction of a default rectgrid (2d unit square, 2x2 elements with -1 as outer neighbour indices)
• rectgrid(rgrid) : copy-constructor
• rectgrid(params) : generate rectgrid with certain options. In this case either
  1. the field grid_initfile is existent in params. Then the file is read and a pointlist p and a triangle. Procedure is then executing the following constructor
  2. a structured equidistant triangle grid is generated
     • params.xnumintervals : number of elements along x directions
     • params.ynumintervals : number of elements along y directions
     • params.xrange,yrange : intervals covered along the axes
     with the optional fields
     • params.bnd_rect_corner1 : coordinates of lower corner of to be marked boundaries
     • params.bnd_rect_corner2 : coordinates of upper corner of to be marked boundaries
     • params.bnd_rect_index: integer index to be set on the edges in the above defined rectangle. Should not be positive integer in the range of the number of elements. use negative indices for certain later discrimination.
     For the last three optional boundary settings, also multiple rectangles can be straightforwardly defined by accepting matrix of columnwise corners1, corners2 and a vector of indices for the different rectangles.

perhaps later: constructor by duneDGF-file? perhaps later: contructor-flag: full vs non-full => only compute redundant information if required.

Note

for diffusion-discretization with FV-schemes, points \(S_i\) must exist, such that \(S_i S_j\) is perpendicular to edge i j, the intersection points are denoted \(S_{ij}\)

Bernard Haasdonk 9.5.2007

Parameters

varargin

variable number of input arguments, see above for description of possible configurations.

Generated fields of grid:

• nelements —  number of elements
• nvertices —  number of vertices
• nneigh —  4
• A —  vector of element area
• Ainv —  vector of inverted element area
• X —  vector of vertex x-coordinates
• Y —  vector of vertex y-coordinates
• VI —  matrix of vertex indices: VI(i,j) is the global index of j-th vertex of element i
• 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) = y-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)
• SY —  vector of y-coordinate of point \(S_j\) (for rect: identical to centroids)
• ESX —  ESX(i,j) = x-coordinate of point \(S_{ij}\) on edge el i to NB j
• ESY —  ESY(i,j) = y-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 \(\alpha \cdot h_i^d \leq A(T_i)\), \(\alpha \cdot \mbox{diam}(T_i) \leq h_i^{d-1}\) and \(\alpha \cdot h_i \leq \)dist(midpoint \(i\) to any neigbour) )

Definition at line 42 of file rectgrid.m.

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Member Function Documentation

function gcopy = rectgrid.copy ( )

virtual

returns a deep copy object of the grid implementation

returns a deep copy object of the grid implementation Every grid implementations needs to provide a deep copy method, because it is derived from a handle class.

Return values

gcopy	Deep copy
gcopy	gcopy

Implements gridbase.

Definition at line 412 of file rectgrid.m.

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function res = rectgrid.isequal

(

grid2

)

function comparing two grids field by field

Parameters

grid2

the grid we want to compare with

Return values

res	boolean indicating whether the two grids are equal

Definition at line 18 of file isequal.m.

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function glob = rectgrid.local2global	(		einds,
			loc,
			params
	)

function performing a local to global coordinate change of vectors of coordinate pairs.

If the first three vertices of a rectangle are v1,v2,v3, then the global coordinate of a single point is

glob = v1 + loc(:,1).*(v2-v1) + loc(:,2).*(v3-v1); 

Parameters

einds	vector of cell indices \(i_k\), \(k=1,...,K\).
loc	matrix of size \(K \times 2\) holding local barycentric coordinate pairs for each cell index \(i_k\), \(k=1,...,K\).
params	params

Return values

glob	global coordinate pairs [X, Y] with vectors X and Y of length \(K\).

Definition at line 18 of file local2global.m.

function p = rectgrid.plot

(

params

)

plot of a rectgrid via plot_polygon_grid()

see help plot_polygon_grid for further information

plot method for a 2D polygonal grid. This routine can be used for triangular and rectangular grids. A line plot is performed as default.

Todo:

For large grids, the routine can be slow. In these cases interestingly, the grid plotting should be implemented with patches, as that seems to be faster...

Parameters

params

optional structure holding fields controlling the plot output.

Return values

p	This is the list of handles to the graphics primitives

Required fields of params:

• axis_tight —  axis tight

Optional fields of params:

• color —  RGB vector of line/patch color
• shrink_factor —  if this flag is given, the elements are plotted shrinked
• plot_patch —  if this flag is set the plot is done by colored patches
• axis_equal —  if this flag is set, set axis to equal scale

Parameters

params

params

Return values

p

p

Definition at line 18 of file plot.m.

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function gridbase grid = rectgrid.set_nbi	(		nbind,
			values
	)

function setting some neighbour indices of a grid to specified values.

Parameters

nbind	neighbour indices
values	new neighbour indices. This can be a single scalar or a vector of the same length as nbind.

Return values

grid

grid

Definition at line 18 of file set_nbi.m.

---

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

• grid/@rectgrid/rectgrid.m
• grid/@rectgrid/demo.m
• grid/@rectgrid/isequal.m
• grid/@rectgrid/local2global.m
• grid/@rectgrid/plot.m
• grid/@rectgrid/set_nbi.m