Detailed Description

Base class for all grid classes.

Definition at line 17 of file gridbase.m.

Inheritance diagram for gridbase:

Collaboration diagram for gridbase:

Public Member Functions
	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...

virtual function gridbase gcopy =	copy ()
	returns a deep copy object of the grid implementation More...

Public Attributes
	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`

Member Function Documentation

function res = gridbase.check_consistency ( )

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

Todo:: : perhaps later further extensions

Return values

res	boolean value which is `true` if the check succeeds, or `false` if it fails.

Definition at line 18 of file check_consistency.m.

function gridbase gcopy = gridbase.copy ( )

pure virtual

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

Implemented in triagrid, rectgrid, cubegrid, and onedgrid.

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function F = gridbase.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.

Computes quadratures approximating integrals

\[ \int_{e(i_k,j_k)} f(s) ds \]

over edges \(e(i_k, j_k)\) for \(k=1,...,K\), where \(i_k\) are cell indices and \(j_k\) denotes local edge numbers.

See also: edge_quad_point() for construction of quadrature points.

Parameters

elids	vector of length \(K\) of cell indices \(i_k\).
edgeids	vector of length \(K\) of local edge indices \(j_k\).
degree	scalar defining the degree of the quadrature rule.
FF	matrix of size `degree x grid.nedges` holding evaluations of a function \(f:\mathbb{R} \to \mathbb{R}\) in quadrature points. This is usually obtained by a call FF=f(edge_quad_points(elids, edgeids, degree)

Return values

F	vector of length \(K\) holding the quadrature evaluations on edges \(e(i_k, j_k)\).

Definition at line 18 of file edge_quad_eval.m.

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function F = gridbase.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.

Computes quadratures approximating integrals

\[ \frac{1}{|e(i_k,j_k)|} \int_{e(i_k,j_k)} f(s) ds \]

over edges \(e(i_k, j_k)\) for \(k=1,...,K\), where \(i_k\) are cell indices and \(j_k\) denotes local edge numbers.

See also: edge_quad_point() for construction of quadrature points.

Parameters

elids	vector of length \(K\) of cell indices \(i_k\).
edgeids	vector of length \(K\) of local edge indices \(j_k\).
degree	scalar defining the degree of the quadrature rule.
FF	matrix of size `degree x grid.nedges` holding evaluations of a function \(f:\mathbb{R} \to \mathbb{R}\) in quadrature points. This is usually obtained by a call FF=f(edge_quad_points(elids, edgeids, degree)

Return values

F	vector of length \(K\) holding the quadrature evaluations on edges \(e(i_k, j_k)\).

Definition at line 18 of file edge_quad_eval_mean.m.

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function PP = gridbase.edge_quad_points	(	elids,
		edgeids,
		degree
	)

get the evaluation points for a quadrature over edges of the given grid.

Given a list of edges \(\{e(i_k, j_k)\}_{i=1}^K\), where \(i_k\) are cell indices and \(j_k\) denotes local edge numbers, this function computes quadrature points

\[\left( x_k^1, \ldots, x_k^Q \right)\]

for each \(k=1,\ldots, K\).

Parameters

elids	vector of length \(K\) of cell indices \(i_k\).
edgeids	vector of length \(K\) of local edge indices \(j_k\).
degree	scalar defining the degree of the quadrature rule.

Return values

PP	matrix of size \(K \times Q\) holding the quadrature for each edge. This can be used as last argument for either edge_quad_eval() or edge_quad_eval_mean().

Definition at line 18 of file edge_quad_points.m.

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function [ P1 , P2 ] = gridbase.get_edge_points	(	elids,
		edgeids
	)

function extracting edge coordinates from the grid.

Given a grid of dimension \(d\), this function returns the two extremal coordinates of edges \(\{e(i_k, j_k)\}_{k=1}^K\) assuming that the edge is \((d-1)\)-dimensional cube. Here, \(i_k\) are cell indices and \(j_k\) denotes local edge numbers for all \(k=1,...,K\)

Parameters

elids	vector of length \(K\) of cell indices \(i_k\).
edgeids	vector of length \(K\) of local edge indices \(j_k\).

Return values

P1	matrix of size \(K x d\) holding the first extremal coordinate of the edges \(e(i_k, j_k)\).
P2	matrix of size \(K x d\) holding the second extremal coordinate of the edges \(e(i_k, j_k)\).

Definition at line 18 of file get_edge_points.m.

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function ENBI = gridbase.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.

arrangement of cell indices

edges' indices of the main cell by which one gets the "right" neighbours

 Example: edge == 1:
   N   |   NE                        ENBI(:,2) = N  | ENBI(:,5) = NE
 ------|-------   <-- nb_ind(0)     ----------------|-----------------
  main |   E                         ENBI(:,1)      | ENBI(:,4)
 ------|-------   <-- nb_ind(1)     ----------------|-----------------
   S   |   SE                        ENBI(:,3)      | ENBI(:,6)
       ^
      edge

Martin Drohmann 05.03.2008

Parameters

edge	edge
tstep	tstep

Return values

ENBI ENBI

Definition at line 18 of file get_enbi.m.

function gridbase gridp = gridbase.gridpart ( eind )

function extracting a part of a triagrid or rectgrid defined by the given element indices in the vector eind.

Note: The neighbour information of the new resulting boundaries is set to -10; The properties gridbase.hmin, gridbase.alpha and the distance-information in the new boundary elements are simply copied. I.e. these fields do not completely meet the definition in the constructor. They might be chosen slightly different, such that the gridp would be really identical to the result generated by the constructor on the subset of points.

Parameters

eind	vector of cell indices which shall be extracted from the grid.

Return values

gridp the partial grid with extracted cells eind.

Generated fields of gridp:

nelements — nelements
nvertices — nvertices
A — A
Ainv — Ainv
X — X
Y — Y
VI — VI
CX — CX
CY — CY
NBI — NBI
INB — INB
EL — EL
DC — DC
NX — NX
NY — NY
ECX — ECX
ECY — ECY
SX — SX
SY — SY
ESX — ESX
ESY — ESY
DS — DS

Definition at line 18 of file gridpart.m.

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function gridbase.inspect ( params )

function plotting various properties of the current grid.

The following plots are generated

element numbers
the neighbour types and
neighbour indices
cog-coordinates
edge-midpoint-coordinates

Parameters

params structure holding control fields for the functions plot_element_data() and plot_sequence() used by this method.

Definition at line 18 of file inspect.m.

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function p = gridbase.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.

By default a patch plot is performed.

Parameters

data	is assumed to be a vector of length `grid.nelements` with element-wise scalar values
plot_params	parameter structure controlling the output of the plot. For time dependent problems `plot_params` are often passed to plot_element_data() and plot_sequence().

Return values

p	is the list of handles to the graphics primitives

Optional fields of plot_params:

shrink_factor — if this flag is given, the elements are plotted shrinked
axis_equal — if this flag is set, set axis to equal scale
no_lines — if this flag is set, no lines are drawn
show_colorbar — if this flag is set, a colorbar is drawn (default 1)
colorbar_location — string specifying the position of the colorbar, e.g. South,EastOutside (default), etc.
clim — if this 2-vector is set, the colorbar is set to these values
geometry_transformation — type of transformation function that is to be applied on the geometry (default none)
postprocess — function data = f(data,[CX,CY],plot_params) postprocessing the given data before it is plotted.
colormap — user-defined colormap
plot_type — string sepcifying wether a patch (default) or a contour plot is created.
line_width — specifyies the line width of plotted lines, e.g. in a contour plot.
contour_lines — number of contour_lines if plot_type is set to contour. The default is 0 and means that the number is determined automatically.

Definition at line 18 of file plot_element_data.m.

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function gridbase.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.

Parameters

data	data
plot_params	parameter structure controlling the output of the plot. For time dependent problems `plot_params` are often passed to plot_element_data() and plot_sequence().

Definition at line 18 of file plot_element_data_sequence.m.

function p = gridbase.plot_polygon_grid ( params )

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

Definition at line 18 of file plot_polygon_grid.m.

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function p = gridbase.plot_vertex_data	(	data,
		params
	)

plot method for vertex data on a 2D polygonal grid.

For example a \(P1\) functions routine can be used for triangular and rectangular grids. A patch plot is performed as default. In case of rectangular elements, no true bilinear interpolation is performed, but the patch is divided into triangles.

Parameters

data	a vector of length `grid.nvertices` with nodewise scalar values.
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:

plot_mode — plot mode
view — view

Optional fields of params:

shrink_factor — if this flag is given, the elements are plotted shrinked
axis_equal — if this flag is set, set axis to equal scale
no_lines — if this flag is set, no lines are drawn
show_colorbar — if this flag is set, a colorbar is drawn (default 1)
colorbar_location — string specifying the position of the colorbar, e.g. South,EastOutside (default), etc.
clim — if this 2-vector is set, the colorbar is set to these values

Definition at line 18 of file plot_vertex_data.m.

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function gridbase grid = gridbase.set_boundary_types ( params )

function setting the boundary types of a polygonal grid.

For rectgrid's this is already done in the constructor. For triagrid this can be performed explicitly here. Existing boundary settings are painted by rectangles The edges with midpoints within such a rectangle are marked accordingly.

required fields or params

bnd_rect_corner1: coordinates of lower corner of to be marked boundaries. This can also be cell array of coordinates for drawing different rectangles. bnd_rect_corner2: coordinates of upper corner of to be marked boundaries. This can also be cell array of coordinates for drawing different rectangles. 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.

-1 is treated as Dirichlet in the numerics
-2 is treated as Neumann in the numerics

This can also be a vector of the same length as the cell arrays defined for bnd_rect_corner1 and bnd_rect_corner2 in case of different boundary types on the domain.

Parameters

params params

Return values

grid grid

Required fields of params:

bnd_rect_index — bnd rect index
bnd_rect_corner1 — bnd rect corner1
bnd_rect_corner2 — bnd rect corner2

Definition at line 18 of file set_boundary_types.m.

Member Data Documentation

gridbase.SX

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

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 are denoted \(S_ij\)

Definition at line 176 of file gridbase.m.

gridbase.SY

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

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 are denoted \(S_ij\)

Definition at line 189 of file gridbase.m.

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

grid/@gridbase/gridbase.m
grid/@gridbase/check_consistency.m
grid/@gridbase/display.m
grid/@gridbase/edge_quad_eval.m
grid/@gridbase/edge_quad_eval_mean.m
grid/@gridbase/edge_quad_points.m
grid/@gridbase/get_edge_points.m
grid/@gridbase/get_enbi.m
grid/@gridbase/gridpart.m
grid/@gridbase/inspect.m
grid/@gridbase/plot_element_data.m
grid/@gridbase/plot_element_data_sequence.m
grid/@gridbase/plot_polygon_grid.m
grid/@gridbase/plot_vertex_data.m
grid/@gridbase/set_boundary_types.m

Detailed Description

Public Member Functions

Public Attributes

Member Function Documentation

Member Data Documentation