About HOPR

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About the Software

The software aims to generate high-order meshes, which are needed for element-based high order schemes like Discontinuous Galerkin schemes, Spectral Element methods or pFEM. Standard mesh generators typically produce straight-edged element meshes, or at most quadratic, including the mid-point. As long as the geometry of the domain is only planar, linear meshes can also be used for high order schemes. However, if domain boundaries are curved, like for example in most aerodynamic simulations, the boundary must be represented more accurately to benefit from the high order resolution. If straight-edged elements are used in this case, nonphysical effects are likely to occur.

Besides in-build mesh generators for simple test cases, it is possible to read externally generated unstructured and structured meshes with straight-edged elements. Several methods can be applied to curve the boundary faces of the elements (see features), and transfinite mappings are used to curve the volume of the boundary elements. Structured meshes can be used for agglomeration to produce a mesh with fully curved hexahedra.

The software is distributed under GPL V3 or later. It is written in Fortran and links to the open-source libraries:

  • HDF5 parallel IO library
  • LAPACK linear algebra package

The software also includes the Hilbert space-filling curve algorithm from D. Moore.


The software was developed in the Numerics Research Group of Prof. Claus-Dieter Munz at the Institute of Aerodynamics and Gasdynamics at the University Stuttgart. Main authors are Thomas Bolemann, Florian Hindenlang and Prof. Claus-Dieter Munz.

Overview Presentation

An overview presentation was given at the "High-Order Computational Fluid Dynamics Technologies Webinar".


You are free to use and modify HOPR for academical purpose. If you use HOPR for your publications, please cite it as follows:

Florian Hindenlang, Thomas Bolemann and Claus-Dieter Munz
Mesh Curving Techniques for High Order Discontinuous Galerkin Simulations
In collection IDIHOM: Industrialization of High-Order Methods-A Top-Down Approach, p.133-152, 2015, Springer


title = {Mesh Curving Techniques for High Order Discontinuous Galerkin Simulations},
author = {F. Hindenlang and T. Bolemann and C-D. Munz},
year = {2015},
date = {2015-01-01},
booktitle = {IDIHOM: Industrialization of High-Order Methods-A Top-Down Approach},
pages = {133--152},
publisher = {Springer}


In-build grid generator

  • Cartesian grids consisting of an arbitrary number of boxes
  • 1 block curved structured grid with given analytic domain boundaries (for example cylinder)
  • Different stretching functions

Curving of external meshes

  • Read-in unstructured linear meshes in CGNS format (tetrahedra, prisms pyramids, hexahedra)
  • Application of Surface Curving via normal vectors:
    • Normal vector reconstruction at curved surfaces from existing grid
    • Analytical expressions for surface normal vectors
    • Read-in of specific Normal vector file (for example from CAD)
  • Application of Surface Curving via interpolation:
    • Additional read-in of subdivided surface mesh
    • Spectral element information from ICEM mesh generator (high order edge information)
  • Volume curving:
    • Read-in of block-structured meshes in structured CGNS format
    • Agglomeration of elements of each structured block to curved hexahedra

Preprocessing for parallel simulations


  • Generates element connectivity for periodic boundary conditions