FEATool
v1.6
Finite Element Analysis Toolbox

After a model geometry has been defined, a computational grid or mesh must be generated to allow for the finite element discretization. This section describes how to create or import a suitable grid.
Grid mode can be selected by pressing the mode button or corresponding menu option. In grid mode the toolbar buttons allow for grid generation, refinement, selecting and deleting grid cells, and setting subdomain and boundary numberings.
The Grid menu options allows for accessing and specifying the Grid Generation Call... which is a Matlab/Octave function call to support external grid generation tools instead of the default one. Generate Quadrilateral Grid calls the structured quadrilateral grid generation routine (see section quadrilateral grid generation section below). Convert Grid Cells is used to convert between triangular and quadrilateral cells in 2D, and between tetrahedral and hexahedral cells in 3D. The Grid Smoothing menu option performs smoothing steps with either Laplacian or umbrella smoothing and allows for a relaxation input parameter.
The Grid menu also supports the following import and export of grids through external ascii format files
Note that in one dimension (1D) the only available option is to create a line grid.
This sectiond describes the format of the grid data structure that FEATool employs as well as advanced command line (CLI) usage such as manually creating and importing grids.
The grid format used by FEATool is specified in the grid struct with the following fields
Field  Description  Size 

p  Grid point coordinates  (n_sdim, n_points) 
c  Grid cell connectivity  (n_edges_per_cell, n_cells) 
a  Grid cell adjacency  (n_edges_per_cell, n_cells) 
b  Boundary information  (3+n_sdim, n_boundary_points) 
s  Grid cell subdomain list  (1, n_cells) 
The coordinates of the grid points are specified by an array p, with the row number indicating the ith coordinate direction, and column number the corresponding grid point number.
Cell connectivities are given in the c array, which specify which grid points make up each cell. Here the row index gives the local vertex number and the column index the cell number. Moreover, the grid points are numbered counterclockwise in each cell.
Adjacency, meaning pointers to neighboring cells, are given in the a array. Similar to c the row index gives the local edge number (starting with the corresponding grid point in c) and the column index points to the cell number. If the edge is on a boundary the corresponding value in a is 0.
Boundary information is specified in the b array. The cell, edge/face, and boundary numbers are given in the first to third rows. The last n_sdim rows give the outward pointing normals corresponding to a boundary edge (or face in 3D). (Note that in higher dimensions each boundary point is specified as many times as boundary edges intersect a point.)
A list of subdomain numbers for each cell is given in the s vector.
The following code can be used to define a one dimensional grid with 10 uniformly spaced cells on the line (0..1)
grid.p = 0:0.1:1; grid.c = [1:10;2:11]; grid.a = [0:9;2:10 0]; grid.b = [1 10;1 2;1 2;1 1]; grid.s = ones(1,10);
Alternatively one can use the grid utility function linegrid
grid = linegrid( 10, 0, 1 );
The plotgrid function can be used to plot and visualize a grid
plotgrid( grid )
A 2 by 2 rectangular grid on the unit square can be created with the following commands
grid.p = [repmat([0,0.5,1],1,3);0 0 0 0.5 0.5 0.5 1 1 1]; grid.c = [1 2 5 4;2 3 6 5;4 5 8 7;5 6 9 8]'; grid.a = [0 2 3 0;0 0 4 1;1 4 0 0;2 0 0 3]; grid.b = [1 2 2 4 4 3 3 1;1 1 2 2 3 3 4 4;1 1 2 2 3 3 4 4; ... 0 0 1 1 0 0 1 1;1 1 0 0 1 1 0 0]; grid.s = ones(1,4);
The rectgrid function can also be used to create rectangular grids, in this case
grid = rectgrid( 2, 2, [0 1;0 1] );
As before the grid can be plotted, visualizing both grid point and cell numbers, with
plotgrid( grid, 'nodelabels', 'on', 'cellabels', 'on' )
Similarly, the boundaries can be visualized with the function plotbdr
plotbdr( grid )
As FEATool also supports simplex triangular cells in two dimensions a grid consisting of quadrilaterals can easily be converted with the utility function quad2tri
grid = quad2tri( grid )
A more complex example, a grid for a rectangle with a circular hole can be created by first creating geometry objects, applying a formula to construct the geometry shape, and then calling the automatic unstructured grid generation function gridgen
% Geometry definition. xmin = 0; xmax = 1; ymin = 0; ymax = 1; tag1 = 'R1'; gobj1 = gobj_rectangle( xmin, xmax, ymin, ymax, tag1 ); xc = (xmaxxmin)/2; yc = (ymaxymin)/2; r = 0.25; tag2 = 'C1'; gobj2 = gobj_circle( [xc yc], r, tag2 ); geom.objects = { gobj1 gobj2 }; formula = 'R1  C1'; geom = geom_apply_formula( geom, formula ); fea.geom = geom; % Grid generation. hmax = 0.1; fea.grid = gridgen( fea, 'hmax', hmax );
As before the grid can be plotted, visualizing both grid point and cell numbers, with
plotgrid( fea )
FEATool allows importing and exporting finite element grids between FeatFlow, GiD, Gmsh, General Mesh Viewer (GMV), and Triangle formats with the corresponding impexp_feat2d, impexp_feat3d, impexp_gid, impexp_gmsh, impexp_gmv, and impexp_triangle functions. These functions can also be accessed from the Grid and Postprocessing menus of the FEATool GUI.
Due to the easy grid format it is possible to manually import grids from other software. The process essentially consists of first exporting the grid point coordinates, and grid cell connectivity data into separate text files. Then import them into Octave or Matlab, after which they can be reshaped and used by FEATool. The following steps describe the process
Load the two files in Octave/Matlab (here it is assumed they are saved as p.txt and c.txt):
load p.txt load c.txt
Use the function gridadj to create an array that points to neighboring cells for each cell edge
n_sdim = size(p,1); a = gridadj(c,n_sdim);
Create a vector that assigns a subdomain number for each cell. If the geometry should be one single block and thus not split, a unit vector is sufficient
n_c = size(c,2); s = ones(1,n_c);
Use the function gridbdr to create boundary edge information
b = gridbdr(p,c,a);
Create a grid struct to hold all the grid information
grid.p = p; grid.c = c; grid.a = a; grid.s = s; grid.b = b;
Optionally, you can also import the grid into the FEATool GUI by using the Menu option
File > Import > Variables From Main Workspace...
Select the created grid variable to import and press Import. This loads it into the local FEATool memory workspace and can now be accessed from the FEATool Terminal (the bottom command **>>** edit field in the main GUI window). By entering the command
fea.grid = grid;
in the FEATool Terminal command line (not the main Octave/Matlab command window) effectively replaces the current grid with the imported one. Press the Grid mode button to update and show the new grid.
The computational finite element library in FEATool supports FEM shape functions for structured grids (quadrilaterals in 2D and hexahedra in 3D). Although more difficult to generate automatically, structured grids are often computationally superior, allowing for higher accuracy with a smaller number of cells.
In FEATool, structured tensorproduct grids of basic geometric shapes can easily be generated on the command line with the following Octave/Matlab mscript functions
Function  Description 

linegrid  Create a 1D line grid 
circgrid  Create a 2D structured circular grid of quadrilateral cells 
holegrid  Create a 2D rectangular grid with a circular hole 
rectgrid  Create a 2D rectangular grid of quadrilateral cells 
ringgrid  Create a 2D grid of a ring shaped domain 
blockgrid  Create a 3D structured block grid 
cylgrid  Create a 3D structured cylindrical grid 
spheregrid  Create a 3D grid for a spherical domain 
Moreover, the grid utility functions delcells, selcells, gridextrude, gridmerge, gridrevolve, gridrotate, and gridscale can be used to manually modify, transform, and join grids to more complex structures. FEATool  Grid transformation and utility functions
The figure below shows three examples of more complex grids created by using these functions.
Create complex structured grids by combining and using the grid utilities together.
The first cylinder benchmark grid is for example created with the following commands:
grid1 = ringgrid( [0.05 0.06 0.08 0.11 0.15], ... 32, [], [], [0.2;0.2] ); grid2 = holegrid( 8, 1, [0 0.41;0 0.41], 0.15, [0.2;0.2] ); grid2 = gridmerge( grid1, 5:8, grid2, 1:4 ); grid3 = rectgrid( [0.41 0.5 0.7 1 1.4 1.8 2.2], ... 8, [0.41 2.2;0 0.41] ); grid = gridmerge( grid3, 4, grid2, 6 );
And the lower right revolved grid with these commands:
grid = rectgrid(); grid = gridscale( grid, {'1(y>0.5).*(y0.5)' 1} ); grid = delcells( selcells( ... '((x<=0.8).*(x>=0.2)).*(y<=0.2)', grid ), grid ); grid = gridrevolve( grid, 20, 0, 1/4, 2, pi/2, 0 );
The last example with two brackets attached to an Ibeam section is more complex:
grid01 = ringgrid( 1, 20, 0.03, 0.06, [0;0] ); indc01 = selcells( grid01, 'y<=sqrt(eps)' ); grid01 = delcells( grid01, indc01 ); grid02 = holegrid( 5, 1, .06*[1 1;1 1], .03, [0;0] ); indc02 = selcells( grid02, 'y>=sqrt(eps)' ); grid02 = delcells( grid02, indc02 ); grid2d = gridmerge( grid01, [5 6], grid02, [7 8] ); grid1 = gridextrude( grid2d, 1, 0.02 ); grid1 = gridrotate( grid1, pi/2, 1 ); grid2 = grid1; grid1.p(2,:) = grid1.p(2,:) + 0.03; grid2.p(2,:) = grid2.p(2,:)  0.01; x_coord = [ 0.08 linspace(0.06,0.06,6) 0.08 ]; y_coord = [ 0.2 0.15 0.1 0.05 0.03 0.01 ... 0.01 0.03 0.05 0.1 0.15 0.2 ]; grid3 = blockgrid( x_coord, y_coord, 1, ... [0.08 0.08;0.2 0.2;0.08 0.06] ); grid4 = blockgrid( 1, y_coord, 5, ... [0.01 0.01;0.2 0.2;0.18 0.08] ); grid5 = grid3; grid5.p(3,:) = grid5.p(3,:)  0.12; grid = gridmerge( grid1, 9, grid3, 6 ); grid = gridmerge( grid2, 9, grid, 17 ); grid = gridmerge( grid4, 6, grid, 23, 1 ); grid = gridmerge( grid5, 6, grid, 29, 2 );
After, the grids have been created on the command line they can also be imported into the FEATool GUI (by using the File > Import > Variables From Main Workspace option, after which the command fea.grid = grid; needs to be entered at the FEATool command prompt).
Using quadrilateral grid cells are often advantageous to simplex or triangular cells in that they can provide somewhat more accuracy when aligned with geometry features and also tends to require less overall grid cells. Quadrilaterals unfortunately do not tend to allow for easy automatic grid generation although it is possible to subdivide triangles into quads the resulting grids are often of poor quality.
The optional gridgen_quad function was originally designed to align quadrilateral cell edges with immersed interfaces described by distance and level set functions. The algorithm then uses the zero level set contour from the distance functions to align grid cell edges with external geometry object boundaries. Furthermore, the cells are split in a way to treat edge cases such as when and interface segment crosses a cell diagonal.
As well as using the Generate Quadrilateral Grid menu option in the FEATool Gui, gridgen_quad can be called on the Matlab command line just as one would call gridgen.
The following functions are available for working with and processing grids
Function  Description 

gridgen  Unstructured automatic grid simplex generation 
gridgen_quad  Unstructured automatic grid quadrilateral generation 
gridrefine  Refine a grid uniformly 
gridsmooth  Apply smoothing to a grid 
gridextrude  Extrude a grid from 2D to 3D 
gridrevolve  Extrude and revolve a 2D grid to 3D 
gridrotate  Rotate a grid along a specified axis 
gridscale  Apply scaling to a grid 
gridmerge  Merge two grids along boundaries 
quad2tri  Convert a grid of quadrilateral cells to triangular cells 
tri2quad  Convert a grid of triangular cells to quadrilateral cells 
hex2tet  Convert a grid of hexahedral cells to tetrahedral cells 
tet2hex  Convert a grid of tetrahedral cells to hexahedral cells 
impexp_feat2d  Import and export 2D Feat(Flow) grid data 
impexp_feat3d  Import and export 3D Feat(Flow) grid data 
impexp_gid  Import and export GiD grid data 
impexp_gmv  Import and export GMV grid and postprocessing data 
impexp_gmsh  Import Gmsh grid and postprocessing data 
impexp_triangle  Import and export 2D Triangle grid and polygon data 
gridcheck  Check grid for errors 
gridadj  Create grid adjacency information 
gridbdr  Create grid boundary information 
gridvert  Create grid vertex information 
gridedge  Create grid edge information 
gridface  Create grid face information 
selcells  Find cell indices from an expression 
delcells  Delete a selection of cells from a grid 
plotbdr  Plot and visualize boundaries 
plotsubd  Plot and visualize subdomains 
plotgrid  Plot and visualize grid 