MultiFEBE

ME-ST-EL-002 [TUTORIAL]
Cube (BEM model)

Á.G. Vega-Artiles
June 2022

Contents
1 Problem description 1

2 Pre-processing 1
2.1 Gmsh format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.1 GEO file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.2 MSH file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Input data file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Results and discussion 9

3.1 Nodal solutions file (*.nso) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Stress resultant solutions (*.tot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Gmsh results file (*.pos) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1 Problem description
In this second tutorial, a static analysis of an elastic cube is performed using the Boundary Element
Method (BEM), i.e. boundary elements. Figure 1 shows the geometry, required material properties
and boundary conditions, where E is the Young’s modulus, ν is the Poisson’s ratio, L is the square
side length and P is the load. Note that self-weight is not considered. The analytical solution to this
problem is easily obtained, which results in the following displacement and stress fields:

(1 + ν)(1 − 2ν)
u1 = P x1 (1)
E(1 − ν)
u2 = u3 = 0 (2)
σ11 = P (3)
ν
σ22 = σ33 = P (4)
1−ν
σ12 = σ13 = σ23 = 0 (5)

from which it can be checked that tractions ti = σij nj (nj are the components of the outward
unit normal components) obviously verify the boundary conditions. Displacement field is linear along
x1 -direction and constant (null) along x2 and x3 -direction. Also, the only non-null stresses (σ11 , σ22
and σ33 ) are constant. Therefore, numerical solution using linear or higher order elements must recover
the analytical solution.
The problem is solved for L = 1 m, E = 1 N/m2 , ν = 0.25 and P = 1 N/m2 and the results are
the followings:

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 u1 = 0.83x1 N/m2 ; u2 = u3 = 0; σ11 = 1 N/m2 ; σ22 = σ33 = 0.3 N/m2 ; σ12 = σ13 = σ23 = 0.

x2

L
P

E, ν

L x1

L
x3

Figure 1: Problem layout.

2 Pre-processing
Pre-processing in MultiFEBE consists of defining the geometry and mesh of the problem. There are
three ways to do such definition: directly from the input file (mode = 0), from another file in the
native format (mode = 1) or from another file in the Gmsh MSH file format version 2.2 (mode = 2)
[1]. However, it is preferable to use a *.msh file for relatively large geometries, therefore, this is the
syntax explained in the following subsections. Furthermore, as the problem will be solved using only
the BEM formulation, the approaches explained in the previous section will be applied.

2.1 Gmsh format

Gmsh [1, 2] is a finite element mesh generator with its own language. The software allows to create
the geometry in a “bottom-up” manner, going from the most basic elements (points) to the highest
order elements (volumes) or in a “Constructive Solid Geometry” manner (boolean operations) or with
a combination of methodologies. Gmsh also allows to define the so-called “physical groups” which are
unmodifiable entities that gather elementary entities as sole entities, where every entity must have a
unique tag. On the other hand, the file containing all this geometric definition is the *.geo file whereas
the mesh definition is in the *.msh file.

2.1.1 GEO file

The Gmsh language allows to define parameters to use later in the script and comments as every other
programming language.
In this example two parameters are defined: the square side length (L) and the mesh element size
(es). Every point in the geometry can be specified by its coordinates (x y z) and the mesh element

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size to use there with the function “Point” and every straight line of the model is created with the
function “Line” and its initial and final points. Then, every surface is created with the function “Plane
Surface” by setting its perimeter with the function “Line Loop” and the lines used to close the surface.
Furthermore, with the function “Physical Surface” every surface is converted into a single entity or
physical entity with its own name.
It is worth noting that “if an expression defines a new entity, it is enclosed between parentheses
but if an expression refers to a previously defined entity, it is enclosed between braces.” [2]
Finally, the mesh generation, the element order and the *.msh file saving can also be specified with
the expressions “Mesh + mesh dimension”, “SetOrder + element order” and “Save + name.msh”,
respectively.
The resulting *.geo file applied to the problem is the following:
L = 1; // Square side length
es = 1; // Element size

// Front Face
Point (1) = {0, 0, L, es};
Point (2) = {L, 0, L, es};
Point (3) = {L, L, L, es};
Point (4) = {0, L, L, es};
Line (1) = {1, 2};
Line (2) = {2, 3};
Line (3) = {3, 4};
Line (4) = {4, 1};
Line Loop (1) = {1, 2, 3, 4};
Plane Surface (1) = {1};
Physical Surface ("front", 1) = {1};

// Right Face
Point(5) = {L, 0, L, es};
Point(6) = {L, 0, 0, es};
Point (7) = {L, L, 0, es};
Point (8) = {L, L, L, es};
Line (5) = {5, 6};
Line (6) = {6, 7};
Line (7) = {7, 8};
Line (8) = {8, 5};
Line Loop (2) = {5, 6, 7, 8};
Plane Surface (2) = {2};
Physical Surface ("right", 2) = {2};

// Back Face
Point(9) = {L, 0, 0, es};
Point(10) = {0, 0, 0, es};
Point (11) = {0, L, 0, es};
Point (12) = {L, L, 0, es};
Line (9) = {9, 10};
Line (10) = {10, 11};
Line (11) = {11, 12};
Line (12) = {12, 9};
Line Loop (3) = {9, 10, 11, 12};
Plane Surface (3) = {3};
Physical Surface ("back", 3) = {3};

// Left Face

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Point(13) = {0, 0, 0, es};
Point(14) = {0, 0, L, es};
Point (15) = {0, L, L, es};
Point (16) = {0, L, 0, es};
Line (13) = {13, 14};
Line (14) = {14, 15};
Line (15) = {15, 16};
Line (16) = {16, 13};
Line Loop (4) = {13, 14, 15, 16};
Plane Surface (4) = {4};
Physical Surface ("left", 4) = {4};

// Top Face
Point(17) = {0, L, L, es};
Point(18) = {L, L, L, es};
Point (19) = {L, L, 0, es};
Point (20) = {0, L, 0, es};
Line (17) = {17, 18};
Line (18) = {18, 19};
Line (19) = {19, 20};
Line (20) = {20, 17};
Line Loop (5) = {17, 18, 19, 20};
Plane Surface (5) = {5};
Physical Surface ("top", 5) = {5};

// Bottom Face
Point(21) = {0, 0, L, es};
Point(22) = {0, 0, 0, es};
Point (23) = {L, 0, 0, es};
Point (24) = {L, 0, L, es};
Line (21) = {21, 22};
Line (22) = {22, 23};
Line (23) = {23, 24};
Line (24) = {24, 21};
Line Loop (6) = {21, 22, 23, 24};
Plane Surface (6) = {6};
Physical Surface ("bottom", 6) = {6};

// Mesh generation
Mesh 2;
SetOrder 1;
Save "t2.msh";

2.1.2 MSH file

The *.msh file begins with a mandatory section about information of the file (MeshFormat) and fol-
lowing by the other sections. Here, three sections are used: the physical group names (PhysicalName),
the nodes (Nodes) and the elements (Elements).
In the section “PhysicalName”, all the physical entities of the model are defined. The first line
indicates the number of physical entities. Then, one line per physical entity indicating the physical
dimension, the tag and the name.
In the section “Nodes”, all the nodes of the model are defined. The first line indicates the number
of nodes. Then, one line per node indicating the node identifier and its coordinates (x y z).

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 Figure 2: Geometry from the *.geo file.

In the section “Elements”, all the elements of the model are defined. The first line indicates the
number of elements. Then, one line per element indicating:

ˆ Element identifier.

ˆ Type of element.

ˆ Number of auxiliary tags.

ˆ List of tags, where the two first auxiliary tags are mandatory, and the first one corresponds to the
identifier of the physical entity to which the element belongs and the second one is the identifier
of the elementary model entity to which the element belongs. The rest of the tags are optional.

ˆ A list of identifiers corresponding to the nodes of the element.

For example, in this case, an element with the identifiers 1 2 2 1 1 1 2 25 corresponds to:

ˆ 1: element 1.

ˆ 2: type 2 (3-node triangle).

ˆ 2: it has 2 auxiliary tags.

ˆ 1: it belongs to the physical entity 1.

ˆ 1: it belongs to the surface 1.

ˆ 1, 2, 25: it connects the nodes 1, 2 and 25.

Finally, the *.msh file can be obtain in File → Export or by following the instructions specified in
the section GEO file.
The resulting *.msh file applied to the problem is the following:

$MeshFormat
2.2 0 8
$EndMeshFormat

$PhysicalNames
6
2 1 "front"
2 2 "right"
2 3 "back"

5
2 4 "left"
2 5 "top"
2 6 "bottom"
$EndPhysicalNames

$Nodes
30
1 0 0 1
2 1 0 1
3 1 1 1
4 0 1 1
5 1 0 1
6 1 0 0
7 1 1 0
8 1 1 1
9 1 0 0
10 0 0 0
11 0 1 0
12 1 1 0
13 0 0 0
14 0 0 1
15 0 1 1
16 0 1 0
17 0 1 1
18 1 1 1
19 1 1 0
20 0 1 0
21 0 0 1
22 0 0 0
23 1 0 0
24 1 0 1
25 0.5 0.5 1
26 1 0.5 0.5
27 0.5 0.5 0
28 0 0.5 0.5
29 0.5 1 0.5
30 0.5 0 0.5
$EndNodes

$Elements
24
1 2 2 1 1 1 2 25
2 2 2 1 1 1 25 4
3 2 2 1 1 2 3 25
4 2 2 1 1 3 4 25
5 2 2 2 2 5 6 26
6 2 2 2 2 5 26 8
7 2 2 2 2 6 7 26
8 2 2 2 2 7 8 26
9 2 2 3 3 9 10 27
10 2 2 3 3 9 27 12
11 2 2 3 3 10 11 27
12 2 2 3 3 11 12 27
13 2 2 4 4 13 14 28

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14 2 2 4 4 13 28 16
15 2 2 4 4 14 15 28
16 2 2 4 4 15 16 28
17 2 2 5 5 17 18 29
18 2 2 5 5 17 29 20
19 2 2 5 5 18 19 29
20 2 2 5 5 19 20 29
21 2 2 6 6 21 22 30
22 2 2 6 6 21 30 24
23 2 2 6 6 22 23 30
24 2 2 6 6 23 24 30
$EndElements

Figure 3: Mesh from the *.msh file.

2.2 Input data file

Solving in MultiFEBE consists of running the software by specifying several options in the following
sections1 : [problem], [settings], [materials], [boundaries], [regions] and [conditions over the boundaries].
The first part to configurate is the problem definition in the section [problem]. This example is a
3D static mechanical problem.

[problem]
n = 3D
type = mechanics
analysis = static

Next step is to configurate the mesh. In this case, a mesh from Gmsh will be used so that it
is necessary to write the option number 2 and the document name obtained from it in the section
[settings]. However, if the mesh were going to be read from the input file, it would require to write the
sections [nodes], [elements] and [parts] instead.

[settings]
mesh_file_mode = 2 "t2.msh"

As the problem has just one material, the section [materials] will need two lines: a first line for the
number of materials in the model and a second line for the properties such as tag, type, E and ν.
1 See reference manual.

7
[materials]
1
1 elastic_solid E 1. nu 0.25

In the section [boundaries], it is necessary to specify the number of boundaries in the first line and
a line per boundary by indicating the boundary identifier, the identifier of the part that discretizes it,
and finally the boundary class. In this example there are 6 boundaries: boundary 1 is the part 1 of
the mesh, boundary 2 the part 2, boundary 3 the part 3, boundary 4 the part 4, boundary 5 the part
5 and boundary 6 the part 6 and all of them are ordinary boundaries.

[boundaries]
6
1 1 ordinary
2 2 ordinary
3 3 ordinary
4 4 ordinary
5 5 ordinary
6 6 ordinary

The format of the section [regions] consists of a first line indicating the number of regions (1).
Furthermore, for each region there must be a block of data consisting of several lines of data. The first
one is the region identifier and the region class (discretization method) (1 be). As the region is a BE
region, then the second line indicates the number and list of boundaries (6 1 2 3 4 5 6). The third line
defines the material (material 1). Then, for a BE region, the fourth line defines the number and list
of BE body loads (0).

[regions]
1

1 be
6 1 2 3 4 5 6
material 1
0

In the section [conditions over be boundaries], all boundaries will be specify in global coordinates
because they are planar and their normal vectors are parallel to one of the global axes. As a 3D
problem, there are three lines for every boundary: a first line for the x direction, the second one for
the y direction and the third one for the z direction, where the first number of every line indicates the
type of condition (here 0 for displacement and 1 for traction) and the second one its value.

[conditions over be boundaries]

boundary 1: 1 0.
1 0.
0 0.
boundary 2: 1 1.
1 0.
1 0.
boundary 3: 1 0.
1 0.
0 0.
boundary 4: 0 0.
1 0.
1 0.
boundary 5: 1 0.
0 0.
1 0.

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boundary 6: 1 0.
0 0.
1 0.

The whole script applied to the problem is the following:

[problem]
n = 3D
type = mechanics
analysis = static

[settings]
mesh_file_mode = 2 "t2.msh"

[materials]
1
1 elastic_solid E 1. nu 0.25

[boundaries]
6
1 1 ordinary
2 2 ordinary
3 3 ordinary
4 4 ordinary
5 5 ordinary
6 6 ordinary

[regions]
1

1 be
6 1 2 3 4 5 6
material 1
0

[conditions over be boundaries]

boundary 1: 1 0.
1 0.
0 0.

boundary 2: 1 1.
1 0.
1 0.

boundary 3: 1 0.
1 0.
0 0.

boundary 4: 0 0.
1 0.
1 0.

boundary 5: 1 0.
0 0.
1 0.

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boundary 6: 1 0.
0 0.
1 0.

3 Results and discussion

3.1 Nodal solutions file (*.nso)
This output file is similar to the file for the 2D example2 , but with more columns due to the z dimension.

3.2 Stress resultant solutions (*.tot)

This output file is a plain text file containing the resultant displacement, rotation, forces and moments
of the centroid of every boundary element. The file starts with a set of header lines whose first character
is “#” (a comment line in GNU/Linux environments), describing the file and the meaning of the main
data columns.

# Program : multifebe
# Version : 0.6
# File_format : tot
# Specification: 1
# Input_file : t2.dat
# Description :
# Timestamp : 2022-05-24 19:20:59.188

# Columns Description
# 1-3 Region id, class and type.
# 4-6 Boundary id, class and face.
# 7-9 Reference point
# 10-12 Resultant displacements along x1, x2 and x3
# 13-15 Resultant rotations in x1, x2 and x3 directions with respect to the reference point
# 16-18 Resultant forces along x1, x2 and x3
# 19-21 Resultant moments in x1, x2 and x3 directions with respect to the reference point

The file for the current example is the following:

Columns 1-9
1__2__3__4__5__6____7__________________________8__________________________9_______________________
1 1 2 1 1 1 0.0000000000000000E+000 0.0000000000000000E+000 0.0000000000000000E+000
1 1 2 2 1 1 0.0000000000000000E+000 0.0000000000000000E+000 0.0000000000000000E+000
1 1 2 3 1 1 0.0000000000000000E+000 0.0000000000000000E+000 0.0000000000000000E+000
1 1 2 4 1 1 0.0000000000000000E+000 0.0000000000000000E+000 0.0000000000000000E+000
1 1 2 5 1 1 0.0000000000000000E+000 0.0000000000000000E+000 0.0000000000000000E+000
1 1 2 6 1 1 0.0000000000000000E+000 0.0000000000000000E+000 0.0000000000000000E+000

Columns 10-12
10_________________________11_________________________12_________________________
416.6665124818034194E-003 -21.7024668712401838E-009 0.0000000000000000E+000
833.3332552543941674E-003 -33.4323049126867974E-009 89.2479633341829967E-009
416.6667054423257577E-003 40.8574092343643430E-009 0.0000000000000000E+000
0.0000000000000000E+000 2.2524860476320668E-009 2.9071411468239618E-009
416.6666696645033863E-003 0.0000000000000000E+000 44.0662267443331586E-009
2 See Tutorial 1.

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 416.6666093799233361E-003 0.0000000000000000E+000 -25.5497694805328682E-009

Columns 13-15
13_________________________14_________________________15_________________________
15.2971017006141144E-009 302.9093621814738513E-003 -308.8234562630653990E-003
149.1666871671048083E-009 302.9093607072384509E-003 -302.9094707058440639E-003
0.0000000000000000E+000 0.0000000000000000E+000 -308.8235240258003778E-003
-646.6494794654632685E-012 0.0000000000000000E+000 0.0000000000000000E+000
31.5797144098362004E-009 308.8234975533534765E-003 -302.9094669326845568E-003
0.0000000000000000E+000 308.8235102637851281E-003 0.0000000000000000E+000

Columns 16-18
16_________________________17_________________________18_________________________
0.0000000000000000E+000 0.0000000000000000E+000 333.3332709461450660E-003
1.0000000000000000E+000 0.0000000000000000E+000 0.0000000000000000E+000
0.0000000000000000E+000 0.0000000000000000E+000 -333.3332716617732894E-003
-999.9999479918649792E-003 0.0000000000000000E+000 0.0000000000000000E+000
0.0000000000000000E+000 333.3332477869328514E-003 0.0000000000000000E+000
0.0000000000000000E+000 -333.3332443552787794E-003 0.0000000000000000E+000

Columns 19-21
19_________________________20_________________________21_________________________
166.6666343762666647E-003 -166.6666073691435712E-003 0.0000000000000000E+000
0.0000000000000000E+000 500.0000000000000000E-003 -500.0000000000001110E-003
-166.6666224357595538E-003 166.6666319121406503E-003 0.0000000000000000E+000
0.0000000000000000E+000 -499.9999707659261161E-003 499.9999745497061832E-003
-166.6666205682695856E-003 0.0000000000000000E+000 166.6666103287823275E-003
166.6666316894805655E-003 0.0000000000000000E+000 -166.6666059412922896E-003

3.3 Gmsh results file (*.pos)

This output file is similar to the file for the 2D example3 , but with more columns due to the z dimension.

3 See Tutorial 1.

11
Figure 4: Nodal displacement results from the *.pos file.

Figure 5: Traction results from the *.pos file.

12
Figure 6: Stress tensor results from the *.pos file.

13
References
[1] C. Geuzaine and J.-F. Remacle, “Gmsh: a three-dimensional finite element mesh generator with
built-in pre- and post-processing facilities.” International Journal for Numerical Methods in En-
gineering, Volume 79, Issue 11, pages 1309–1331, (2009).
[2] C. Geuzaine and J.-F. Remacle, “Gmsh.” http://gmsh.info/

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