1. Straight-Edged Boxes

HOPR has several simple built-in mesh generators.

Fig. 1.1 HOPR output: Mesh of multiple cartesian boxes with a stretched element arrangement.

1.1. Cartesian Box

This tutorial shows how to generate a simple mesh of a cubical box and the definition of boundary conditions. The parameter file can be found in

tutorials/1-01-cartbox/parameter.ini

See Cartesian Box: Exemplary Variations of Boundary Conditions for cases with different boundary conditions.

1.1.1. Cartesian Box: Description of Parameters

The following table describes all parameters present in the parameter file. Therefore, it is limited to the parameters needed to generate a Cartesian box mesh. A description of all parameters can be found in List of Parameters.

Table 1.3 Parameters Cartesian Box.

Parameters	Setting	Description
ProjectName	cartbox	Defines the name embedded in the generated mesh file and is used as prefix for the output files (cartbox_mesh.h5, cartbox_Debugmesh.dat, cartbox_Debugmesh_BC.dat, …).
DebugVisu	T	Generation of visualization files for the volume and boundary mesh (*_Debugmesh.dat, *_Debugmesh_BC.dat) to aid debugging.
Mode	1	Mode of mesh generation; 1: Cartesian mesh (build-in), 3: CFD General Notation System (CGNS, extern)
nZones	1	Number of Cartesian boxes
Corner	(/0.,0.,0. ,,1.,0.,0. ,,1.,1.,0. ,,0.,1.,0. ,,0.,0.,1. ,,1.,0.,1. ,,1.,1.,1. ,,0.,1.,1. /)	Coordinates of the box’s corner nodes in the three-dimensional Cartesian coordinate system. Refer to Fig. 4.1 for the node ordering. Furthermore, the corner nodes define the six surfaces of the cartesian box, see Table 4.2.
nElems	(/2,3,4/)	Number of elements per box in the direction of the Cartesian coordinate axes; (/nElemX,nElemY,nElemZ/)
BCIndex	(/1,2,3,4,5,6/)	The BCIndex parameter assigns a bondary condition to each surface of the cartesian box in order of the surfaces. The number of a vector’s component represents the nth boundary condition in order of its position in the file. Hence, each position refers to the six box sides (/z-,y-,x+,y+,x-,z+/). Indices can be applied repeatedly, see Cartesian Box: Exemplary Variations of Boundary Conditions for details.
ElemType	108	Type of cells/elements used for discretization; 104: Tetrahedron, 105: Pyramid, 106: Prism with triangular base, 108: Hexahedron
BoundaryName	BC_zminus	Name of each boundary condition
BoundaryType	(/4,0,0,0/)	Type of each boundary condition, provided with the components (/ Type, curveIndex, State, alpha /).For a single Cartesian box, only the component Type has to be set. The other components are set to the default value 0. Further description of the components can be found in the next tutorials or in the List of Parameters.

1.1.2. Cartesian Box: Boundary Conditions and Sketch

Fig. 1.2 shows the sketch of the current flow setup. Fig. 1.3 gives the corresponding excerpt of the parameter file containing the boundary conditions. Entries are colored to highlight the connection between the BCIndex and the boundary conditions. The same colors are used for the visualization below.

Fig. 1.2 Sketch of the current flow setup

Fig. 1.3 Cartesian Box Boundary Conditions ini-File

The index of the first component of the BCIndex vector 1 assigns the boundary condition on position one, BC_zminus, to the first surface. Consequently, the index of the second component of the BCIndex vector 2 assigns the second boundary condition BC_yminus to the second surface of the Cartesian box and so on. More examples are given in Cartesian Box: Exemplary Variations of Boundary Conditions.

1.1.3. Cartesian Box: Output Visualization

The following section highlights the visualization of the above setup. For more details on the HOPR visualization output, see Visualization.

Mesh

This is a visualization of the cartbox_Debugmesh.dat file.

Fig. 1.4 Mesh of the cartesian box

Boundary Conditions

This is a visualization of the cartbox_Debugmesh_BC.dat file. The colors of the surfaces represent the boundary conditions and are identical to the ones in the excerpt of the parameter file.

Fig. 1.5 Boundary condition 1 (BC_wall) is assigned to surface 1

Fig. 1.6 Boundary condition 2 (BC_inflow) is assigned to surface 2

Fig. 1.7 Boundary condition 3 (BC_outflow) is assigned to surface 3

Fig. 1.8 Boundary condition 4 (BC_yplus) is assigned to surface 4

Fig. 1.9 Boundary condition 5 (BC_xminus) is assigned to surface 5

Fig. 1.10 Boundary condition 6 (BC_zplus) is assigned to surface 6

1.1.4. Cartesian Box: Exemplary Variations of Boundary Conditions

For a better understanding of the interaction between the parameter BCIndex and the declaration of the boundary conditions, two different examples are presented below. Both examples are slight variations of the tutorial Cartesian Box in that way, that there are only three different boundary conditions: WALL, INFLOW and OUTFLOW. This means the components belonging to the surfaces will be equal in the BCIndex vector (see List of Parameters).

1.1.4.1. Example 1

Parameters and Sketch

The parameter file of this example can be found in

/tutorials/1-01-cartbox/parameter_ex1.ini

Fig. 1.11 shows a sketch of the current problem. Figure 1.12 shows a snippet of the parameter file which deals with the boundary conditions. In this code snippet, some text elements are colored to show the connection between the surfaces and their associated boundary conditions. The same colors are used for the visualization below.

The first four components of the BCIndex vector are equal. The index of these components 1 means that the first boundary condition, BC_wall, is assigned to the surfaces one to four. Furthermore, the fifth component of the BCIndex vector with the index 2 means that the second boundary condition BC_inflow is assigned to the fifth surface of the Cartesian box. The third boundary condition BC_outflow is assigned to the the sixth surface. Therefore, the last or the sixth component of the BCIndex vector is set to 3.

Fig. 1.11 Sketch of Example 1

Fig. 1.12 Cartesian Box example 1 Boundary Conditions ini-File

Output Vizualisation

If you need help visualizing the HOPR output, visit the Visualization page.

Mesh

A visualization of the mesh of the cartbox_ex1_Debugmesh.dat file is given below.

Fig. 1.13 Mesh of the cartesian box

Boundary Conditions

This is a visualization of the cartbox_ex1_Debugmesh_BC.dat file. The colors of the surfaces represent the boundary conditions and are the same as in the snippet of the parameter file.

Fig. 1.14 Boundary condition 1 (BC_wall) is assigned to surface 1-4

Fig. 1.15 Boundary condition 2 (BC_inflow) is assigned to surface 5

Fig. 1.16 Boundary condition 3 (BC_outflow) is assigned to surface 6

1.1.4.2. Example 2

The parameter file of this example can be found in

/tutorials/1-01-cartbox/parameter_ex2.ini

Figure 1.17 shows the sketch of the current problem. Figure 1.18 shows a snippet of the parameter file which deals with the boundary conditions. In this code snippet, some text elements are colored to show the connection between the surfaces and their asociated boundary conditions. The same colors are used for the visualization below.

In this example the first, third, and sixth component of the BCIndex vector are equal. The index of these components 1 means that the boundary condition at position one, BC_wall, is assigned to the surfaces one, three and six. Furthermore, the fifth component of the BCIndex vector with the index 2 means that the second boundary condition BC_inflow is assigned to the fifth surface of the cartesian box. The third boundary condition BC_outflow is assigned to the second and the forth surface. Therefore, the second and fourth component of the BCIndex vector are set to 3.

Parameters and Sketch

Fig. 1.17 Sketch of Example 2

Fig. 1.18 Cartesian Box example 2 Boundary Conditions ini-File

Output Vizualisation

If you need help visualizing the HOPR output, visit the Visualization page.

Mesh

A visualization of the mesh of the cartbox_ex2_Debugmesh.dat file is given below.

Fig. 1.19 Mesh of the cartesian box

Boundary Conditions

This is a visualization of the cartbox_ex2_Debugmesh_BC.dat file. The colors of the surfaces represent the boundary conditions and are the same as in the snippet of the parameter file.

Fig. 1.20 Boundary condition 1 (BC_wall) is assigned to surfaces 1, 3 and 6

Fig. 1.21 Boundary condition 2 (BC_inflow) is assigned to surface 5

Fig. 1.22 Boundary condition 3 (BC_outflow) is assigned to surface 2 and 4

1.2. Periodic Boundary Conditions

This tutorial shows how to define periodic boundary conditions, using only slightly modified definitions from the Cartesian Box tutorial. The parameter file can be found in

tutorials/1-02-cartbox_periodic/parameter.ini

1.2.1. Periodic Boundary Conditions: Description of Parameters

In the following parameters of the parameter file are explained. This description consists only of parameters which are necessary to generate a periodic boundary condition. A description of all parameters of the parameterfile can be found in the previous tutorial Cartesian Box and List of Parameters.

Table 1.4 Parameters Periodic Boundary Conditions.

Parameters	Setting	Description
BoundaryName	BC_zminus	Name of the boundary condition
BoundaryType	(/1,0,0,-1/)	For each periodic boundary condition three parameters are mandatory, the BoundaryName, the BoundaryType and the displacement vector vv. The Type parameter consists of four components to set: (/ Type, curveIndex, State, alpha /). For a periodic boundary condition the component Type has always to set to “1”. The fourth component alpha assigns a displacement vector vv and its direction (-/+) to the periodic boundary. An alpha of “-1” means that the first (“1”) defined displacement vector is assigned to this surface in the opposite direction (“-1”) as he was defined. For a simple cartesian box the other components curveIndex and State have to be set 0. The further description of the components can be found in the next tutorials or in the List of Parameters.
vv	(/0.,0.,1./)	The displacement vector must be specified in the three-dimensional Cartesian coordinate system and must be normal to a surface to which the vector was assigned. In addition, the displacement vector must point to the inside of the Cartesian box. In case of two parallel surface planes, both with periodic boundary conditions, only one displacement vector needs to be defined. Therefore, the different directions of the vectors can be compensated by changing the sign of alpha, the fourth component of the BoundaryType vector. It has to be taken into account that the displacement vector has to be as long as the distance between the surfaces to which it is assigned. The index of a displacement vector is also defined by the position of its definition, as the parameter BCIndex. Multiple boundary condition definitions between two definitions of displacement vectors do not affect the index of the displacement vectors.

1.2.2. Periodic Boundary Conditions: Boundary Conditions and Sketch

Figure 1.23 shows the sketch of the current problem. It is similar to the problem in the Cartesian Box tutorial, but instead of Dirichlet, periodic boundary conditions are assigned to the surfaces one, two, four, and six. Below is a code snippet of the parameter file that deals with the periodic boundary conditions. In this code snippet, some text elements are colored to show the connection between boundary conditions and their related displacement vectors. The same colors are used for the visualization in Fig. 1.17.

Fig. 1.23 Sketch of the current problem; For a greater clarity in this figure the displacement vectors are shown shorter than they are. In truth the vector arrows are as long as the side length of the cartesian box.

As one can see, the first four boundary conditions are periodic because the last alpha components of the BoundaryType parameters are not equal to zero. In the definitions of the first two boundary conditions which are assigned to the surfaces one and six (see BCindex) the alpha component is set to 1. This means that the assiciated displacement vector is the first defined displacement vector in the parameter file. The sign of alpha can be explained by the position of the associated surfaces. On the one hand, the vector must point to the inside of the Cartesian box. On the other hand, the vector must also point to the other surfaces that have been assigned with the periodic boundary condition. The vector itself must point in the direction of the z-axis because it must be normal to the surface. In addition, the side length of the Cartesian box is one, so all defined displacement vectors have a length of one. For the other two periodic boundary conditions of surfaces two and four, the second defined displacement vector is consulted (see alpha value of the BoundaryType parameters). The components of the displacement vector (/0.,1.,0./) result from the required perpendicularity to the surfaces and the side length of the Cartesian box.

Fig. 1.24 Cartbox periodic ini-File.

1.3. Multiple Cartesian Boxes

This tutorial shows how to create a mesh consisting of multiple Cartesian boxes and how to link them via boundary conditions. The parameter file can be found in

tutorials/1-03-cartbox_multiple/parameter.ini

1.3.1. Multiple Cartesian Boxes: Definition of Multiple Cartesian Boxes

The following general snippet of the parameter file shows how to define multiple Cartesian boxes in the parameter file.

!====================================================================== !
! MESH
!====================================================================== !
  Mode         =1                       ! Mode for Cartesian boxes
  nZones       =n                       ! number of boxes

! ===  zone 1 ===
  ...

! ===  zone 2  ===
  ...

  .
  .
  .

! ===  zone n  ===
 ...

First, the parameter nZones must be adapted to the number of Cartesian boxes to be defined. The Cartesian boxes can be defined simply by writing them and their specifications (Corner, nElems, BCIndex, elemtype, ...) among themselves. Furthermore, it is important that the boxes are correctly defined to each other. If there is to be a contact between two boxes, it is mandatory that the corresponding surfaces will coincide. This means that the corner nodes of the surfaces must also coincide.

However, a correct definition of the corner nodes is not sufficient for the functionality. Therefore, the parameter BCIndex, which assigns boundary conditions to the surfaces of the boxes must be adapted.

Table 1.5 Multiple Cartesian Boxes.

Parameters	Setting	Description
BCIndex	(/0,0,0,0,0,0/)	The BCIndex parameter assigns a boundary condition to each surface of the Cartesian box in the order of the surfaces. The number of the component of a vector represents the nth boundary condition in order of its position in the file. Thus, each position refers to the six sides of the box (/z-,y-,x+,y+,x-,z+/). In case of multiple Cartesian boxes, there are surfaces that coincide with other surfaces. No boundary condition can be assigned to such surfaces. Therefore, the number of the corresponding vector’s component is set to 0. Here, all components of the parameter BCIndex are set to 0 ((/0,0,0,0,0,0/)). This means that this box is surrounded by six other boxes, so that no boundary condition can be assigned to a single surface.

A description of all parameters of the parameter file can be found in List of Parameters.

1.3.2. Multiple Cartesian Boxes: Sketch

Figure 1.25 shows the sketch of the current problem. As one can see, the generated mesh shall consist of three Cartesian boxes. These zones are defined in the parameter file in the following order:

1st zone: lower left zone

            BCIndex setting: (/-,-,-,-,-,0/)

2nd zone: upper left zone

            BCIndex setting: (/0,-,-,-,-,-/)

3rd zone: upper right zone

            BCIndex setting: (/-,-,-,-,0,-/)

For a better understanding, the different settings of the parameter BCIndex are also given. The given settings only consider the components that are set to 0 due to coinciding surfaces.

Fig. 1.25 Sketch of the current problem; for clarity, the displacement vectors are shown shorter than they are. In reality, the vector arrows are as long as the sides of the Cartesian box.

1.3.3. Multiple Cartesian Boxes: Output Visualization

If you need help visualizing the HOPR output, visit the Visualization page.

Mesh

A visualization of the mesh of the cartbox_multiple_Debugmesh.dat file is given below.

Fig. 1.26 Mesh of the multiple Cartesian boxes.

Boundary Conditions

A visualization of the mesh of the cartbox_multiple_Debugmesh_BC.dat file is given below. The colors of the surfaces represent the boundary conditions and are the same as in the snippet of the parameter file.

Fig. 1.27 Boundary condition 1 (BC_slipwall) is assigned to surface 1 of the first box	Fig. 1.28 Boundary condition 2 (BC_upperwall) is assigned to surface 6 of the second and the third box	Fig. 1.29 Boundary condition 3 (BC_lowerwall) is assigned to surface 3 of the first box and to surface 1 of the third box
Fig. 1.30 Boundary condition 4 (BC_inflow) is assigned to surface 5 of the first and the second box	Fig. 1.31 Boundary condition 5 (BC_outflow) is assigned to surface 3 of the third box	Fig. 1.32 Periodic boundary condition 6 (BC_yminus) is assigned to surface 2 of the first, the second and the third box
Fig. 1.33 Periodic boundary condition 7 (BC_yplus) is assigned to surface 4 of the first, the second and the third box

1.4. Stretching Functions

This tutorial shows how to create a mesh consisting of a boxes with a stretched element arrangement. The parameter file can be found in

tutorials/1-04-cartbox_multiple_stretch/parameter.ini

1.4.1. Stretching Functions: Definition of Stretching Functions

Stretching functions can be used to generate a mesh consisting of boxes with a stretched element arrangement. Therefore, two new parameters can be defined in the parameter file: factor and l0. Each of them can be used to stretch the elements of a box. Their meaning and connection is shown as one-dimensional case in Fig. 1.28.

Fig. 1.34 "Stretch parameters" factor and l0

In case of a stretched element arrangement, the next element of a box is always stretched by a factor \(f\) in the direction of the coordinate axis. The value of factor \(f\) can have a positive or a negative sign. Here \(f\) has either a negative and an absolute value >1 or a positive sign and an absolute value <1. The length of the first element is called \(l_{0}\). All streched elements \(N\) together have the length \(l_{ges}\).

1.4.2. Stretching Functions: Building a Cartesian Box with Stretched Elements

To obtain a single Cartesian box with a stretched element arrangement, it is important to know that the parameters \(l_{0}\) and \(N\) are defined even before the stretching parameters factor and l0 are defined. The length \(l_{ges}\) is defined by the boundaries of the Cartesian box and the number of elements per box in the direction of the Cartesian coordinate axes, hereafter called \(N\), is defined by the parameter nElems.

For a stretched element arrangement either the parameter factor or the parameter l0 must be defined. The other missing parameter is calculated internally with the following equation:

\[ \frac{l_{ges}}{l_{0}} = \sum_{i=1}^{N} f^{i-1} = \frac{1 - f^{N}}{1 - f } \]

The structure of the two parameters is explained below. A description of all parameters in the parameter file can be found in List of Parameters.

Table 1.6 Stretching Functions.

Parameters	Setting	Description
factor	(/-1.75,1,-1.5/)	Stretching factor of the elements in the direction of the Cartesian coordinate axes. A value >1 means an increase of the element size in the direction of the coordinate axis, while a value in the intervall (0,1) means a decrease. A value of 1 has no effect on the element sizes, and a value of 0 disables the stretching function for this axis. Furthermore, the stretching behavior can be mirrored by adding a negative sign to the values. A combination with the parameter l0 ignores the element number of the defined box. In case of (/-1.75,1,-1.5/) each following element is compressed by a factor of 1.75 in the direction of the x-axis and 1.5 in the direction of the z-axis. The element arrangement along the y-axis is not changed.
l0	(/0,1,5,0/)	The length of the first element of a stretched element arrangement of a Cartesian box. Each component of the vector represents an axis of the Cartesian coordinate system. A value of 0 disables the stretching function for that axis. A negative sign defines the length of the first element of the other side of the box. A combination with the parameter factor ignores the element number of the defined box. Here, the stretching function is disabled for the x- and z-axis, while the first element in the direction of the y-axis has a size of 1.5.

Variable Definition of Stretching Functions

There are several ways to get a stretched element arrangement due to the two different stretching parameters factor and l0. These cases are:

• Definition of factor: In this case, the next element of a box is stretched by a factor \(f\) in the direction of the coordinate axis. The parameter l0 is calculated internally by the equation provided above. The number of elements \(N\), defined by the parameter nElems, remains unchanged.

• Definition of l0: In this case, the length of the first element (or of the last, if the sign is negative) in the direction of the coordinate axis is defined. The parameter factor is calculated internally by the equation provided above. The number of elements \(N\), defined by the parameter nElems remains unchanged.

• Definition of factor and l0: In this case, the parameters must be defined manually by the equation provided above. Otherwise, the stretched element arrangement will most likely not achieved the desired shape. In case of an inaccurate definition, the parameter factor will ne adjusted to the parameter l0 which means that factor will changed internally. Furthermore, the number of elements \(N\), defined by the parameter nElems will probably not be kept. Instead, \(N\) will be rounded to nearest natural number.

These three different cases are illustrated below using a small cube with an edge length of one and with four elements per axis. For a better understanding, only the x- and y- values have been changed and visualized.

Fig. 1.35 Non-stretched element arrangement nElems =(/4,4,4/) factor =(/0,0,0/) l0 =(/0,0,0/)

Fig. 1.36 Stretched element arrangement. The element size in the direction of the x-axis increases by a factor of 1.5. In the direction of the y-axis it decreases by the factor of 1.2. nElems =(/4,4,4/) factor =(/1.5,-1.2,0/) l0 =(/0,0,0/)

Fig. 1.37 Stretched element arrangement. The first element in the direction of the x-axis has a length of 0.5 and a length of 0.2 in the direction of the y-axis. The parameter factor is adjusted. nElems =(/4,4,4/) factor =(/0,0,0/) l0 =(/0.5,0.2,0/)

Fig. 1.38 Stretched element arrangement with a combination of factor and l0. The parameter l0 defines the side lengths of the first element. The following element sizes are multiplied by the compontens of the parameter factor. The number of elements \(N\) which is defined by the parameter nElems is not kept. nElems =(/4,4,4/) factor =(/1.5,-1.2,0/) l0 =(/0.5,0.2,0/)

1.4.3. Stretching Functions: Building Multiple Cartesian Boxes with Stretched Elements

To create a mesh consisting of multiple Cartesian boxes with a stretched element arrangement at least one of the parameters factor and l0 must be defined for each Cartesian box. The reason for this is that if there is to be a contact between two boxes, the corner nodes of the surfaces have to coincide. This means that defining a stretch function for one Cartesian box leads to the need of a stretch function for the neighboring Cartesian box. A visualization of this issue can be seen in the sketch of the tutorial’s problem.

1.4.4. Stretching Functions: Sketch

The following is an example of a mesh of multiple Cartesian boxes with a stretched element arrangement. The corresponding parameter file can be found in

tutorials/1-04-cartbox_multiple_stretch/parameter.ini

The arrangement of the Cartesian boxes is the same as in the Multiple Cartesian Boxes tutorial but instead of equidistant elements, a stretched element arrangement is created by inserting the parameters factor and l0. Furthermore, the number of elements of each box has been increased by changing the parameter nElems to better visualize the stretched element arrangement. The figure below shows how the elements are stretched.

Fig. 1.39 Sketch of a mesh with multiple Cartesian boxes with a stretched element arrangement. The arrows indicate the direction in which the elements are compressed. The elements become smaller the closer they are to the Cartesian x-y-plane and y-z-plane. The elements in the direction of the Cartesian y-axis remain equidistant.

Note that the figure above only shows in which directions the elements are stretched or compressed. Neither the number of elements per box nor the Cartesian box sizes shown correspond to the parameters of the parameter file.

1.4.5. Stretching Functions: Output Visualization

If you need help visualizing the HOPR output, visit the Visualization page. A visualization of the mesh of the cartbox_multiple_stretch_mesh.h5 file is given below.

Fig. 1.40 Side view of the mesh of the multiple cartesian boxes with a stretched element arrangement.

Fig. 1.41 Mesh of the multiple cartesian boxes with a stretched element arrangement.