Internal Mesh Generator¶

A simple internal mesh generator for generating curved meshes of basic block-structured geometries

The internal generator creates structured, block-wise meshes (Cartesian boxes) with options for multi-zone setups, periodic couplings, and graded (stretched) element distributions.

Mesh Specification¶

Internal meshes are specified using exclusively the INI parameter file format. As internal meshes are created using zones with 8 corner nodes, each zone is limited to a Cartesian box geometry (before mesh post-processing). The following minimal parameter file can be found in tutorials/1-01-cartbox and generates a simple Cartesian box mesh.

!================================================================================================================================= !
! OUTPUT
!================================================================================================================================= !
ProjectName  = 1-01-cartbox                         ! Name of output files
OutputFormat = HDF5                                 ! Mesh output format [HDF5 VTK GMSH]

!================================================================================================================================= !
! MESH
!================================================================================================================================= !
Mode         = 1                                    ! Mode for Cartesian boxes
nZones       = 1                                    ! Number of 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. /)
                                                    ! Corner node positions: (/ x_1,y_1,z_1, x_2,y_2,z_2,..... , x_8,y_8,z_8/)
nElems       = (/8,8,8/)                            ! Number of elements in each direction
BCIndex      = (/1,2,3,4,5,6/)                      ! Indices of boundary conditions for six boundary faces (z-,y-,x+,y+,x-,z+)
ElemType     = 108                                  ! Element type (104: Tetrahedra, 105: Pyramid, 106: Prism, 108: Hexahedral)
nGeo         = 9

!================================================================================================================================= !
! BOUNDARY CONDITIONS
!================================================================================================================================= !
BoundaryName = BC_zminus                            ! BC index 1 (from position in parameterfile)
BoundaryType = (/4,0,0,0/)                          ! (/ Type, curveIndex, State, alpha /)
BoundaryName = BC_yminus                            ! BC index 2
BoundaryType = (/2,0,0,0/)
BoundaryName = BC_xplus                             ! BC index 3
BoundaryType = (/2,0,0,0/)
BoundaryName = BC_yplus                             ! BC index 4
BoundaryType = (/2,0,0,0/)
BoundaryName = BC_xminus                            ! BC index 5
BoundaryType = (/2,0,0,0/)
BoundaryName = BC_zplus                             ! BC index 6
BoundaryType = (/9,0,0,0/)

Info

Mesh output in VTK / Gmsh format is provided mainly for visualization purposes and might not support all high-order features.

Multiple Cartesian Boxes¶

PyHOPE supports multi-zone setups consisting of multiple Cartesian boxes. Each box, refered to as zone, is defined by its 8 corner nodes specified in the Corner vector. The number of elements in each direction for each box is specified in the nElems vector. If boxes are connected with each other, the corner nodes of the touching element faces must either be identical or be placed on the center of the element edge (non-conforming interfaces with hanging nodes).

Boundary Conditions¶

Boundary conditions are specified in the parameter file using the BoundaryName and BoundaryType keywords. The order of the boundary conditions is determined by the order of the boundary faces given as z-,y-,x+,y+,x-,z+ inside the BCIndex vector. Indices can be applied repeatedly.

Periodic Boundary Conditions¶

Boundary faces can be coupled periodically by specifying the BoundaryType = (/1,0,0,-1/) where the last entry alpha specifies the periodic displacement vector to be used. A negative alpha corresponds to a displacement vector applied in the opposite direction. Displacement vectors are indexed in the order of appearance in parameter file. The following minimal parameter file can be found in tutorials/1-02-cartbox-periodic and generates a Cartesian box mesh with periodic coupling in x- and y-direction.

!================================================================================================================================= !
! OUTPUT
!================================================================================================================================= !
ProjectName  = 1-02-cartbox_periodic                ! Name of output files
Debugvisu    = F                                    ! Launch the GMSH GUI to visualize the mesh
OutputFormat = HDF5                                 ! Mesh output format (HDF5 VTK)

!================================================================================================================================= !
! MESH
!================================================================================================================================= !
Mode         = 1                                    ! Mode for Cartesian boxes
nZones       = 1                                    ! Number of 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. /)
                                                    ! Corner node positions: (/ x_1,y_1,z_1, x_2,y_2,z_2,..... , x_8,y_8,z_8/)
nElems       = (/2,3,4/)                            ! Number of elements in each direction
BCIndex      = (/1,3,6,4,5,2/)                      ! Indices of boundary conditions for six boundary faces (z-,y-,x+,y+,x-,z+)
ElemType     = 108                                  ! Element type (108: Hexahedral)
doFEMConnect = T                                    ! Generate finite element method (FEM) connectivity

!================================================================================================================================= !
! BOUNDARY CONDITIONS
!================================================================================================================================= !
BoundaryName = BC_zminus                            ! BC index 1 (from position in parameterfile)
BoundaryType = (/1,0,0,1/)                          ! (/ Type, curveIndex, State, alpha /)
BoundaryName = BC_zplus                             ! BC index 2
BoundaryType = (/1,0,0,-1/)                         ! here the direction of the vector 1 is changed, because it is the opposite side
vv = (/0.,0.,1./)                                   ! vector for periodic BC in z direction (zminus,zplus), index=1

BoundaryName = BC_yminus                            ! BC index 3
BoundaryType = (/1,0,0,2/)
BoundaryName = BC_yplus                             ! BC index 4
BoundaryType = (/1,0,0,-2/)                         ! (/ BCType=1: periodic, 0, 0, Index of second vector vv in parameter file /)
vv = (/0.,1.,0./)                                   ! vector for periodic BC in y direction (yminus,yplus), index=2

BoundaryName = BC_inflow                            ! BC index 5
BoundaryType = (/2,0,0,0/)
BoundaryName = BC_outflow                           ! BC index 6
BoundaryType = (/2,0,0,0/)

Stretching Functions¶

By default, elements are distributed uniformly in each direction within each transfinite mesh zone. Stretching (grading) of the element distribution used to generate a mesh consisting of boxes with a stretched element arrangement. The following parameters control the stretching and are defined for each zone separately.

StrechType:: Stretching type. Possible values are 0 (no stretching, uniform element distribution), 1 (stretching with progression factor), 2 (stretching with a length ratio) and 3 (stretching with a bell function).
factor:: Stretching factor. Values > 1.0 lead to an exponential increase of the element size in the respective direction, values < 1.0 lead to an exponential decrease of the element size in the respective direction. 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.
l0:: Length scale. 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.
DXmaxToDXmin:: Frame ratio. The ratio of the largest to the smallest element size in the respective direction. If the stretchType vector component for an axis is 3, the element arrangement is controlled by DXmaxToDXmin instead of the parameter fac.

Calculation Formulas¶

The stretching functions are applied in each direction independently. The following formulas are used to calculate the element sizes and positions.

Calculation of the element size for stretchType=1:

\[ {\Delta x_{i+1} = f \cdot \Delta x_{i} f = \left(\frac{\Delta x_{max}}{\Delta x_{min}} \right)^{1/(nElems - 1)} } \]

Calculation of the element size for stretchType=3:

\[ \Delta x(\xi) \sim 1 + \left( \frac{\Delta x_{max}}{\Delta x_{min}}-1\right)\cdot \left( \frac{\exp[-(\xi \cdot f)^2] - \exp[-f^2]}{\exp[0] - \exp[-f^2]}\right) \]

Post-Deformation¶

Mesh zones can be deformed after the initial mesh generation using a set of pre-defined deformation functions. The resulting curved multi-block mesh is composed of of several structured boxes, which are first assembled and then globally mapped to a curved domain.

convtest:: Convergence test mapping. A simple sinusoidal deformation in all directions to test the convergence behavior of high-order methods on curved meshes.
cylinder:: A cylindrical mapping. The 2D square region is mapped to a cylindrical or toroidal coordinate system.
sphere:: A spherical mapping. The 3D cubic region is mapped to a spherical coordinate system inside \([-1, 1]^3\).
phill:: Periodic hill mapping. A 2D mapping to generate an extruded periodic hill geometry.

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