FEATool
v1.7 Finite Element Analysis Toolbox |

Grid

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.

- calls the default grid generation function to generate and unstructured grid of triangles in 2D and tetrahedra in 3D. The desired overall mean
**Grid Size**is specified in the corresponding edit field. - The button is used to uniformly refine a grid. This will split triangular and quadrilateral (2D) grid cells into four subcells, and tetrahedral and hexahedral (3D) ones into eight subcells each. Newly created grid points on boundaries will also be aligned with any geometries in 2D.
- Grid cells can be selected pressing the button and entering a logical expression in the corresponding dialog box.
*x*,*y*, and*z*can be used in the expression to represent the space dimension coordinates, for example*x>0.5*(note that due to limited floating point precision and numerical inaccuracies it is sometimes better to perform logical tests such as these with a small tolerance, for example*x>=0.5-1e-6*).

- After a selection has been made the selected cells can be deleted by pressing the button.
- Similarly the subdomain number of the selected cells can be changed by pressing the button. The initial subdomain number shown in the
**Set Subdomain**dialog box corresponds to the*current number of subdomains + 1*, assuming that a completely new subdomain is to be created from the selected cells.

- The button allows changing and modifying assignment of boundary numbering. The
**Edit Boundaries**dialog box allows one to call the automatic boundary calculation routine which tries to split boundary cells according to the angle between boundary edges and faces. Alternatively, one can manually select boundary cells, edges, and faces by entering a logical expression such as*x>0&y>0*and*ib==2*which here selects the boundaries in the top positive quadrant and with current boundary group equal to 2, respectively.

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

**Import Grid > FeatFlow Format...**imports a grid in FEAT / FeatFlow format (.tri/.prm files) (calls impexp_feat2d or impexp_feat3d).**Import Grid > FEniCS/Dolfin Format...**imports a grid in FEniCS/Dolfin format (.xml file) (calls impexp_dolfin).**Import Grid > GiD Format...**imports a grid in GiD format (.msh file) (calls impexp_gid).**Import Grid > Gmsh Format...**imports a grid in Gmsh format (.msh file) (calls impexp_gmsh).**Import Grid > GMV Format...**imports a grid in General Mesh Viewer (GMV) format (.gmv file) (calls impexp_gmv).**Import Grid > Triangle Format...**imports a 2D Triangle grid (.node/.ele files) (calls impexp_triangle).

**Export Grid > FeatFlow Format...**exports a grid in FEAT / FeatFlow (.tri/.prm files) (calls impexp_feat2d or impexp_feat3d).**Export Grid > FEniCS/Dolfin Format...**exports a grid in FEniCS/Dolfin format (.xml file) (calls impexp_dolfin).**Export Grid > GiD Format...**exports a grid in GiD format (.msh file) (calls impexp_gid).**Export Grid > Gmsh Format...**exports a grid in Gmsh format (.msh file) (calls impexp_gmsh).**Export Grid > GMV Format...**exports a grid in General Mesh Viewer (GMV) format (.gmv file) (calls impexp_gmv).**Export Grid > Triangle Format...**exports a 2D Triangle polygonal geometry description (.poly file) (calls impexp_triangle).

Note that in one dimension (1D) the only available option is to create a line grid.

This section 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 *i*-th coordinate direction, and column number the corresponding grid point number.

Cell connectivities are specified in the **c** array, where each column point to the grid points (in **p**) making up each cell. 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 specified in the **a** array. Similar to **c** the row index gives the local edge or face number (starting with the corresponding grid point in **c**) and the column index points to the cell number. If the edge is on an external boundary the corresponding entry in **a** is *0*.

Boundary information is specified in the **b** array. The cell, edge/face, and boundary numbers are prescribed in the first to third rows. The last *n_sdim* rows consists of the outward pointing unit normals corresponding to each boundary cell edge (or face in 3D).

A list of subdomain numbers for each cell is given in the **s** vector.

The various grid cells available in FEATool are defined in this sub section.

In one dimension only the simple straight line segment grid cell is available. Grid Example 1 shows how this can be defined and used.

The 2D triangular grid cell is defined with vertices in counter clockwise order. Local edges e_{i} are defined by the local vertices v_{i} as

[ e1 ; [ v1 v2 ; e2 ; = v2 v3 ; e3 ] v3 v1 ];

The quadrilateral grid cell is similarly defined with vertices in counter clockwise order. Local edges e_{i} are defined by the local vertices v_{i} as

[ e1 ; [ v1 v2 ; e2 ; v2 v3 ; e3 ] = v3 v4 ; e4 ] v4 v1 ];

The 3D tetrahedral grid cell is defined with vertices in counter clockwise order. Local edges e_{i} are defined by the local vertices v_{i} as

[ e1 ; [ v1 v2 ; e2 ; v2 v3 ; e3 ; v3 v1 ; e4 ; = v1 v4 ; e5 ; v2 v4 ; e6 ] v3 v4 ];

The local faces f_{i} are defined by the local vertices v_{i} as

[ f1 ; [ v1 v2 v3 ; f2 ; v1 v2 v4 ; f3 ; = v2 v3 v4 ; f4 ] v3 v1 v4 ];

The hexahedral brick grid cell is defined with vertices in counter clockwise order. Local edges e_{i} are defined by the local vertices v_{i} as

[ e1 ; [ v1 v2 ; e2 ; v2 v3 ; e3 ; v3 v4 ; e4 ; v4 v1 ; e5 ; v1 v5 ; e6 ; v2 v6 ; e7 ; = v3 v7 ; e8 ; v4 v8 ; e9 ; v5 v6 ; e10 ; v6 v7 ; e11 ; v7 v8 ; e12 ] v8 v1 ];

The local faces f_{i} are defined by the local vertices v_{i} as

[ f1 ; [ v1 v2 v3 v4 ; f2 ; v1 v2 v6 v5 ; f3 ; v2 v3 v7 v6 ; f4 ; = v3 v4 v8 v7 ; f5 ; v4 v1 v5 v8 ; f6 ] v5 v6 v7 v8 ];

The following code can be used to manually 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 unit square rectangular grid 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 (subdomains can be visualized with plotsubd)

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 )

Reverse conversions can be made with tri2quad function which uses Catmull-Clark subdivision. In 3D the hex2tet and tet2hex functions perform similar conversions.

A more complex example, a grid for a rectangle with a circular hole can be created by first creating geometry objects (a rectangle and a circle), 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 = (xmax-xmin)/2; yc = (ymax-ymin)/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( grid, 'nodelabels', 'on', 'cellabels', 'on' )

FEATool allows importing and exporting finite element grids between FeatFlow, FEniCS, GiD, Gmsh, General Mesh Viewer (GMV), and Triangle formats with the corresponding impexp_feat2d, impexp_feat3d, impexp_dolfin, 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 simple grid format syntax 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 from the external grid generation tool 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

- The first step is to output the grid to an ASCII text format. If possible save the grid output to two files, one for the grid vertex coordinates, and another with the grid cell connectivities (this specifies which grid points/vertices make up each cell). If this is not possible one will have to manually open the grid output file in a text editor, and cut and save the grid coordinates and cell connectivities to two different files.
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

- Reshape the grid coordinates (
**p**variable) so that it has the form where is the number of space dimensions (1, 2, or 3) and is the total number of grid points (**p**essentially consists of rows of x, y, and z-coordinates). Precisely how to reshape depends on the output format from the external grid generator, one might not have to do anything (check the shape by entering the command**size(p)**or**whos**), it might be enough to transpose as**p = p';**, or one might have to reshape with something like**p = reshape(p,n_sdim,n_p);**. - Similarly, reshape the cell connectivity array
**c**so that it has the shape where is the number of vertices on each cell (for example 3 for triangles) and is the total number of cells. Each column should point to the corresponding grid points in**p**that make up the cell. - The grid vertices must run in counterclockwise order on each cell, so reorient them if necessary. This is usually accomplished by changing the row order, for example
**c = c([3 2 1],:);**which the ordering for triangles. 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 row vector is sufficient

n_c = size(c,2); s = ones( 1, n_c );

Use the function gridbdr to create boundary edge/face information. The

*gridbdr*function splits external boundaries where the angles are acuteb = gridbdr( p, c, a );

Alternatively, one can manually construct and set the boundary numbering, for example as

[ix_edge_face,ix_cells] = find( a==0 ); b = [ix_cells'; ix_edge_face'; ones(1,length(ix_cells))]; b = [b; gridnormals(p,c,b)];

this creates a boundary array

**b**where all external boundaries are joined. With additional boundary information one can now split the boundary edges and faces (columns) by assigning different boundary numbers (third row in**b**). A helpful function to use here is findbdr which allows one to find boundary numbers and column indices in to**b**by prescribing logical expressions such as*x>0.5*. For example[~,ix] = findbdr( struct('p',p,'c',c,'b',b), 'x>0.5', false ); b(3,ix) = 2;

sets boundary number

*2*on all boundary edges/faces where*x>0.5*is true.Create a grid

*struct*to hold all the grid informationgrid.p = p; grid.c = c; grid.a = a; grid.s = s; grid.b = b;

- Now the grid can be used by FEATool functions and subroutines on the command line.
Optionally, one 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 commandfea.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 tensor-product grids of basic geometric shapes can easily be generated on the command line with the following Octave/Matlab m-script 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 selcells, delcells, gridextrude, gridmerge, gridrevolve, gridrotate, and gridscale can be used to manually modify, transform, and join grids to more complex structures.

The figure below shows three examples of more complex grids created by using these functions.

The first flow over 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).*(y-0.5)' 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 I-beam 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 an 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_dolfin | Import and export FEniCS/Dolfin 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 |

gridbdrx | Reconstruct internal boundaries |

gridvert | Create grid vertex information |

gridedge | Create grid edge information |

gridface | Create grid face information |

findbdr | Find boundary indices and numbers |

sort_bdr | Sort boundaries according to geometry objects |

project_bdrp | Project boundary points to geometric boundary |

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 |